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\section{1~~~Why do we need SUSY ?%
  \label{why-do-we-need-susy}%
}


\subsection{1.1~~~Drawbacks and unsolved problems of Standard Model%
  \label{drawbacks-and-unsolved-problems-of-standard-model}%
}

Despite of Standard Model (SM) tremendous success, the theory
is not believed to be theoretically satisfactory, and is
regarded as low-energy effective theory of a more fundamental
theory. There are many reasons for the conceptual incompleteness:
the assignment of the quantum numbers to the fermions
is not fully clear; why  whitin SM      the bulk matter is neutral;
we don't understand why there are three apparantly unrelated
gauge groups and what rules the strength of their
coupling constants; there is no a sufficient reason for
which there are only  three generations of fermions; it's not explained
why the fermion mass spectrum ranges over thirteen orders of magnitude.
Moreover, the presence of the scalar field in SM is completely
artifical, since it's introduced just for the purpose of breaking
the electroweak symmetry. There is only one scalar boson while all other bosons are vectors in the theory.
The problem of CP-violation described by \DUrole{raw-tex}{$\theta_{QCD}$} phase   is not well understood including CP-violation in a strong
interaction. Another question is related to flavour mixing and the number of generations which are arbitrary.
In the strict framework of the SM, nuetrinos are massless.
However, there is an experimental evidence that neutrinos
must be massive particles, although very light ones.

The Standard Model depends on nineteen parameters: the three gauge coupling constants, the two parameters \DUrole{raw-tex}{$\mu^{2}$} and
\DUrole{raw-tex}{$\lambda$} which determine the mass and the self coupling of the
Higgs field, the nine quark and charged lepton masses, the three
angles and one phase specifying the quark mixing matrix, the
\DUrole{raw-tex}{$\theta_{QCD}$} phase related to strong spontaneous  CP violation \DUrole{raw-tex}{\cite{Dine:2000cj},\cite{Kim:2008hd}}
\DUrole{raw-tex}{$$\frac{\theta_{QCD}}{32\pi^2}g_S^2\tilde{G}^{a,\mu\nu}G_{\mu\nu}^a,$$}
where \DUrole{raw-tex}{$ G^{a}_{\mu\nu}$} is the strength tensor of gluon field and
\DUrole{raw-tex}{$\tilde{G}^{a}_{\mu\nu}$} is the axial strength tensor \DUrole{raw-tex}{$$\tilde{G}^{a}_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}G^{a,\alpha\beta}. $$}
Moreover, many more parameters are needed to accommodate non-accelerator observations \DUrole{raw-tex}{\cite{Ashie:2004m}, \cite{Ahmad:2002jz}}.
The results from the Super Kamiokande experiment on the leptons observed in the atmospheric showers of particles stimulated by cosmic rays incident on the top of the atmosphere seem
to clearly indicate that the muon neutrino exhibits oscillatory behavior. This can interpreter neutrinos as Majorana particles
requiring  three neutrino masses, three mixing angles and one CP-violating phase.
Most physicists believe that this number of absolutely arbitrary parameters  is just too much for the fundamental theory.

Gravity is not fundamentally unied with the other interactions in the Standard
Model, although it is possible to graft on classical general relativity by
hand. However, general relativity is not a quantum theory, and there is no
obvious way to generate one within the standard model context.
Possible solutions include supergravity theories \DUrole{raw-tex}{\cite{Nilles:1983ge}}.
In addition to the fact that gravity is not unied and not quantized
there is another dificulty, namely the non-zero cosmological constant \DUrole{raw-tex}{$\Lambda_{cosm}$}.  The \emph{fine-tunning} of
primordial cosmological constant \DUrole{raw-tex}{$\Lambda_{bare}$},
\DUrole{raw-tex}{$$\Lambda_{cosm}=\Lambda_{bare} + \Lambda_{SSB},$$} \DUrole{raw-tex}{$$ \Lambda_{SSB}=8{\pi}G_N<0|V|0>\sim 10^{56}\Lambda_{obs} $$}
which can be thought of as the value of the energy of the vacuum in the absence of
spontaneous symmetry breaking, is required if someone  couple the electrowek interaction to the gravity.

Another question is: can all the interactions be unified?
Radiative effects make gauge couplings dependent on the energy scale.
The couplings, when defined as renormalized values including loop corrections, require the
specification of a renormalization prescription for which the modified minimal subtraction (MS)
scheme \DUrole{raw-tex}{\cite{tHooft:1973mm},\cite{Bardeen:1978yd}} is used.
In the SM the strong and weak couplings associated with non-Abelian gauge groups decrease
with energy, as it's shown for \DUrole{raw-tex}{$\alpha_S$} of strong interaction
in Figure \DUrole{raw-tex}{\ref{fig1}}, while the electromagnetic one associated with the Abelian group on the contrary
increases. Thus, it becomes possible that at some energy scale they become equal. According
to the Grand Unification Theory's (GUT) idea, this equality is not occasional but is a manifestation of a unique origin of
these three interactions. As a result of spontaneous symmetry breaking, the unifying group is
broken and the unique interaction is splitted into three branches which are called strong, weak and
electromagnetic interactions. Figure \DUrole{raw-tex}{\ref{fig2}} (left plot) clearly demonstrates that within the SM the coupling constant unification
at a single point is impossible. This result means that the unification can only be obtained if new physics enters between
the electroweak and the Planck scales, \DUrole{raw-tex}{$M_{P}=\sqrt{\hbar c^5/G_N} \sim 10^{19}$} GeV where Newton constant \DUrole{raw-tex}{$G_N$} is
extremely small \DUrole{raw-tex}{$G_N \sim 10 ^{-38}$} GeV\textsuperscript{-2}.  It turns out
that within the SUSY model a perfect unification can be obtained if the SUSY
masses are of an order of 1 TeV. This is shown in Figure \DUrole{raw-tex}{$\ref{fig2}$} (right plot).
From the fit requiring unification, one finds values, shown in Equation \DUrole{raw-tex}{\ref{eq1}}
for the break point \DUrole{raw-tex}{$M_{SUSY}$} and the unification point \DUrole{raw-tex}{$M_{GUT}$}. The unification is too
perfect in a supersymmetric theories.

\begin{flalign}\label{eq1}
M_{SUSY}   = 10^{3.4\pm 0.9\pm 0.4} GeV, \nonumber \\
M_{GUT}  = 10^{15.8\pm 0.3\pm 0.1} GeV, \nonumber \\
\alpha^{-1}_{GUT}  = 26.3 \pm 1.9 \pm 1.0
\end{flalign}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/running_alphaS.PNG}}
\caption{Summary of running of the strong coupling \DUrole{raw-tex}{$\alpha_{S}$\cite{Bethke:2000ai}\label{fig1}}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/GUT-SM-MSSM-3couplings.PNG}}
\caption{Evolution of the inverse of the three coupling constants in the Standard Model (left) and in the supersymmetric extension of the SM (MSSM) (right). Only in the latter case
unification is obtained. The SUSY particles are assumed to contribute only above the effective SUSY scale \DUrole{raw-tex}{$M_{SUSY}$} of about 1 TeV, which causes a change in the slope in the evolution of couplings. The thickness of the lines represents the error in the coupling constants. \DUrole{raw-tex}{\label{fig2}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Another solution for GUT is \DUrole{raw-tex}{SU(5)}. In the minimal \DUrole{raw-tex}{SU(5)} model \DUrole{raw-tex}{\cite{1974PhRvL..32..438G}}, interactions almost unify at
\DUrole{raw-tex}{$10^{16}GeV$}.  It's the simplest group which
has rank 4 (dimension of Cartan sub-algebra, SM has rank 4 as well ).
The matter fields (fermions and leptons) can be fitted well to  \DUrole{raw-tex}{$\bf\bar{5}$}  fundamental represanation and
\DUrole{raw-tex}{$\bf\bar{10}$}  representation by \DUrole{raw-tex}{5x5} antisymmetric matrices. Altogether, these are 15 degrees of freedom, just like in
one generation of SM. The unification within \DUrole{raw-tex}{SU(5)} results in only one coupling, quantization of the electric
charge \emph{Q} (the hypercharge is now a traceless generator \DUrole{raw-tex}{$Y$},   \DUrole{raw-tex}{$TrY=0$}), non-conserving barion and lepton numbers which
allows decays of protons with \DUrole{raw-tex}{$\tau \sim 10^{30} - 10^{31}$\cite{1978NuPhB.135...66B}}

To get the desired spontaneous symmetry breaking pattern in GUT, one needs two different scales \DUrole{raw-tex}{$V$}
and  \DUrole{raw-tex}{$v$} in a GUT , namely, \DUrole{raw-tex}{$M_W$} and \DUrole{raw-tex}{$M_{GUT}$} ,what leads
to a very serious problem which is called the hierarchy problem.

\begin{flalign}\label{eq2}
M_{H} \sim M_{W} \sim v \sim 10^2\, GeV, \nonumber \\
M_{GUT} \sim M_{\Xi}\sim V \sim 10^{16}\, GeV, \nonumber \\
M_{W}/M_{GUT}\sim 10^{-14},
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$H$} and \DUrole{raw-tex}{$\Xi$} are the scalar fields (light and heavy Higgs bosons )
responsible for the spontaneous breaking of SU(2) and GUT groups, respectively.
The question arises of how to get so small number \DUrole{raw-tex}{$M_{W}/M_{GUT}$} in a natural way. One needs some kind
of fine tuning in a theory, and we don’t know if there anything behind it.

Another aspect of the hierarchy problem is  the preservation of a given
hierarchy by \DUrole{raw-tex}{(\ref{eq2})}. The radiative corrections will destroy it.
To see this, consider the radiative correction to the light Higgs boson mass \DUrole{raw-tex}{$M_{H}$}. It is given by the Feynman
diagrams shown in Figure \DUrole{raw-tex}{\ref{fig3}}. The diagrams give the  ultra-violet (UV) quadratic divergency \DUrole{raw-tex}{(\ref{eq3})} at most.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Feynmann_comb.pdf}}
\caption{Tadpole and self-energy Feynmann diagrams for Higgs boson \DUrole{raw-tex}{\label{fig3}}. Here \DUrole{raw-tex}{$V$} stands for gauge bosons, \DUrole{raw-tex}{$e_k$} denotes
fermions. \DUrole{raw-tex}{\label{fig3}}}
\end{figure}

\begin{flalign}\label{eq3}
\delta M_{H}^2 \sim \lambda^2 \int \frac{d^4k}{(2\pi)^4}\frac{i}{k^2-M_H^2} ~ \lambda^2 \Lambda^2
    \end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$\lambda$} is a Yukawa coupling, \DUrole{raw-tex}{$\Lambda$} is  cut-off scale much larger than \DUrole{raw-tex}{$1\, TeV$},  perhaps of order of the Plank scale \DUrole{raw-tex}{$M_p$} or \DUrole{raw-tex}{$M_{GUT}$}.

The divergency spoils the hierarchy if it's not cancelled. This very accurate cancellation
with a precision \DUrole{raw-tex}{$\sim 10^{-14}$} needs a fine tuning of the coupling constants.
In the Standard Model, the  Higgs mass suffer from the quadratic divergent  correction, while the photon \DUrole{raw-tex}{$m_{\gamma}$} and
fermions \DUrole{raw-tex}{$m_f$} masses
are protected, because of  gauge  invariance and chiral symmetry, respectively.

\begin{flalign}\label{eq3}
\delta m_{\gamma}^2 \sim e^2 q^2 Log\Lambda^2,  \nonumber \\
\delta m_{f}^2 \sim e^2 m_f^2 Log\Lambda^2,
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$q$} is the photon momentum.

The only known way of achieving this kind of cancellation of quadratic terms (also known as
the cancellation of the quadratic divergencies) is supersymmetry (SUSY). Moreover, SUSY automatically
cancels quadratic corrections in all orders of Pertubation Theory (PT). This is due to the contributions of superpartners
of ordinary particles. The superparners  are originated   due to the invariance of the theory on supersymmetric
transformation. The contribution from boson loops cancels those from the fermion ones.
because of an additional factor (-1) coming from Fermi statistics, as shown in Figure \DUrole{raw-tex}{\ref{fig5}.}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Feynmann_SUSY_hierarcy_solving.pdf}}
\caption{Cancellation of quadratic divergency \DUrole{raw-tex}{\label{fig5}}.}
\end{figure}

The top diagrams, shown in Figure \DUrole{raw-tex}{\ref{fig5}}, are contributions of the Higgs bosons and its superpartner. The
strength is given by Yukawa coupling \DUrole{raw-tex}{$\lambda$}. The botom digrams of
Figure \DUrole{raw-tex}{\ref{fig5}} represents the gauge interaction of Higgs with gauge  bosons and gauginos
which is proportional to the gauge coupling \DUrole{raw-tex}{$g$}. The full cancellation takes place
in the case of unbroken supersymmetry when the sum rule \DUrole{raw-tex}{(\ref{eq4})} relating masses of bosons and their superpartners
is valid

\begin{flalign}\label{eq4}
\sum_{bosons} m^2 = \sum_{fermions} m^2
\end{flalign}

\DUrole{raw-tex}{\noindent}
If the equation \DUrole{raw-tex}{(\ref{eq4})} violated then SUSY is broken, and cancellation is true up to the SUSY breaking scale,
\DUrole{raw-tex}{$M_{SUSY}$}, given by \DUrole{raw-tex}{(\ref{eq5})}

\begin{flalign}\label{eq5}
\sum_{bosons} m^2 - \sum_{fermions} m^2  = M_{SUSY}^2
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$M_{SUSY}$} should not be very large, \DUrole{raw-tex}{$M_{SUSY}\leq 1\, TeV$,} to make the \DUrole{raw-tex}{$fine-tuning$} natural, as a
consequence, the radiative correction will be of order of Higgs boson mass

\begin{flalign}\label{eq6}
\delta M_{H} \sim g^2 \cdot M_{SUSY}^2 \sim 10^{-1\cdot 2} \cdot 10^{3\cdot 2} \sim 10^{2\cdot 2} \sim M_H^2.
\end{flalign}


\section{2~~~Basics of supersymmetry%
  \label{basics-of-supersymmetry}%
}


\subsection{2.1~~~Superspace and Super-Poincare Lie Algebra%
  \label{superspace-and-super-poincare-lie-algebra}%
}

I start the introduction  to the supersymmetry by  explaining the superspace, superfileds  and
giving an  algebra of supersymmetric transformation,
\DUrole{raw-tex}{Super-Poincare Lie} Algebra, which contains additional SUSY generators \DUrole{raw-tex}{$Q_\alpha^i$} and \DUrole{raw-tex}{$\bar{Q}_{\alpha}^i$}.
Here  \DUrole{raw-tex}{$\alpha$} and \DUrole{raw-tex}{$i$} are  spinorial  and supersymmetric indexes. I'll illustrate the ideas of
supersymmetry using  a simple example having only one supersymmetric dimension \DUrole{raw-tex}{$i=1$} which is
usually  denoted as
SUSY with only one chiral supermultiplet, \DUrole{raw-tex}{$N=1$}. The model was first written by Wees and Zumino \DUrole{raw-tex}{\cite{Wess:1974jb}}
The spinorial SUSY charge \DUrole{raw-tex}{$Q_{\alpha}^i$} performs transformations of the matter fields,
when supertranslation in the superspace is done. The superspace \DUrole{raw-tex}{\cite{}} differs from the ordinary
Euclidean (Minkowski) space by  adding of two new coordinates, \DUrole{raw-tex}{$\theta_{\alpha}$} and \DUrole{raw-tex}{$\bar{\theta}_{\alpha}$}
which are Grassmanian, ie anticommuting, variables \DUrole{raw-tex}{($\ref{eq7}$)},
where \DUrole{raw-tex}{$\bar{\theta}$} variable is obtained from conjugating
\DUrole{raw-tex}{$\theta$}

\begin{flalign}\label{eq7}
\left\{\theta_{\alpha},\theta_{\beta}\right\} = \left\{\bar{\theta}_{\alpha},\bar{\theta}_{\beta}\right\}=0,     \nonumber \\
\left\{\frac{\partial}{\partial \theta_{\alpha}},\theta_{\beta}\right\} =  \left\{\frac{\partial}{\partial \bar{\theta}_{\alpha}},\bar{\theta}_{\beta}\right\} = \delta_{\alpha\beta}, \nonumber \\
\left\{\frac{\partial}{\partial \theta_{\alpha}},\bar{\theta}_{\beta}\right\} =  \left\{\frac{\partial}{\partial \bar{\theta}_{\alpha}},\theta_{\beta}\right\} = 0,
\end{flalign}

\DUrole{raw-tex}{\noindent}
The Minkowski space  transforms to superspace, as shown in \DUrole{raw-tex}{($\ref{eq8}$)}

\begin{flalign}\label{eq8}
{x_\mu} \rightarrow  {x_\mu,\theta,\bar{\theta}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
A SUSY group element \DUrole{raw-tex}{$(\ref{eq9})$} in the representation applicable for Weyl spinors and scalar fields
can be constructed in  the superspace in the same way as an ordinary
translation in the usual space

\begin{flalign}\label{eq9}
G(x_\mu,\theta_{\alpha},\bar{\theta}_{\dot{\alpha}}) = e^i\cdot(-x^{\mu}P_{\mu}+\theta_{\alpha}Q_{\alpha}+\bar{\theta}_{\dot{\alpha}}\bar{Q}_{\dot{\alpha}})
\end{flalign}

\DUrole{raw-tex}{\noindent}
I'll skip spinorial indexes in further calculations assuming the following notation for the representations of
scalars, pseudoscalars and 4-vectors constructed on Grassman variables

\DUrole{raw-tex}{$$\theta\theta \rightarrow \epsilon^{\alpha\beta}\theta_{\alpha}\theta_{\beta}, \\
\bar{\theta}\bar{\theta} \rightarrow \epsilon^{\dot{\alpha}\dot{\beta}}\bar{\theta}_{\dot{\alpha}}\bar{\theta}_{\dot{\beta}}, \\
\bar{\theta}\sigma^{\mu}\theta \rightarrow \bar{\theta}^{\dot{\alpha}}\sigma^{\mu}_{\dot{\alpha}\alpha}\theta^{\alpha},
$$}

where \DUrole{raw-tex}{$\sigma^{\mu}$} \DUrole{raw-tex}{\cite{}} defined by  \DUrole{raw-tex}{$2\times2$} identity matrix \DUrole{raw-tex}{$I_{2\times2}$} and three vector of
Pauli matrices  \DUrole{raw-tex}{$\vec{\sigma}=(\sigma^1,\sigma^2,\sigma^3)$} \DUrole{raw-tex}{\cite{}} \DUrole{raw-tex}{$$\sigma^{\mu}=(I_{2x2},\vec{\sigma})$$}
and \DUrole{raw-tex}{$\epsilon^{\alpha\beta}$} is the antisymmetric tensor \DUrole{raw-tex}{\cite{}}
Any infinitesimal  trnasformations in the superspace induced by \DUrole{raw-tex}{$(\ref{eq9})$} take the form \DUrole{raw-tex}{$(\ref{eq10})$}

\begin{flalign}\label{eq10}
 x_{\mu} \rightarrow  x_{\mu} + i\theta\sigma_{\mu}\bar{\epsilon} + i\epsilon\sigma_{\mu}\bar{\theta}, \nonumber \\
 \theta \rightarrow  \theta + \epsilon, \nonumber \\
 \bar{\theta} \rightarrow  \bar{\theta} + \bar{\epsilon}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can deduce the expressions of \DUrole{raw-tex}{$Q$} and \DUrole{raw-tex}{$\bar{Q}$}  SUSY generators from \DUrole{raw-tex}{$(\ref{eq9})$} and \DUrole{raw-tex}{$(\ref{eq10})$}

 \begin{flalign}\label{eq11}
  Q = \frac{\partial}{\partial \theta} - i \bar{\theta}\sigma_{\mu}\partial_{\mu}, \nonumber \\
  \bar{Q} = -\frac{\partial}{\partial \bar{\theta}} + i \sigma_{\mu}\theta\partial_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
It's obvious to show the main rule of the graded (Super-Poinkare )  Lie Algebra \DUrole{raw-tex}{\cite{}}. This is  the
anticommutator of  \DUrole{raw-tex}{$Q$} and \DUrole{raw-tex}{$\bar{Q}$}.

 \begin{flalign}\label{eq11}
 \{Q_\alpha,\bar{Q}_\beta\} = 2 \sigma^{\mu}_{\alpha\beta}P_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The presence of the translation generator in \DUrole{raw-tex}{$(\ref{eq11})$} shows that the supersymmetry is a spacetime
symmetry. It means that the supersymmetry is conserved in time \DUrole{raw-tex}{$(\ref{eq12})$}

\begin{flalign}\label{eq12}
[Q_{\alpha},P^0] = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
More general case of  the equtation \DUrole{raw-tex}{$(\ref{eq12})$}  can be written as \DUrole{raw-tex}{$(\ref{eq13})$}

\begin{flalign}\label{eq13}
[Q_{\alpha},P^{\mu}] = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The equation \DUrole{raw-tex}{$(\ref{eq13})$}  is the second rule of the graded Lie Algebra.
The commutators of \DUrole{raw-tex}{$Q(\bar{Q})$} with the Lorentz group generators \DUrole{raw-tex}{$M_{\mu\nu}$} of  the angular
momentum are fixed because the supersymmetric charge was declared to be a spin \DUrole{raw-tex}{$1/2$} Weyl spinor, i.e \DUrole{raw-tex}{$Q(\bar{Q})\sim \psi_{L(R)}$}.
The commutator  becomes

\begin{flalign}\label{eq14}
[Q_\alpha,M^{\mu\nu}] = 1/2\sigma^{\mu\nu}_{\alpha\beta}Q_{\beta}=-[\bar{Q}_{\alpha},M_{\mu\nu}] , \nonumber\\
 \sigma^{\mu\nu}_{\alpha\beta}=\frac{i}{4}[\sigma_{\mu},\sigma_{\nu}]
\end{flalign}

\DUrole{raw-tex}{\noindent}
Models with more than one SUSY charge, \DUrole{raw-tex}{$Q_{\alpha}$} (extended SUSY) in the low energy theory do not lead to chiral fermions and so
are excluded for phenomenological reasons.

We immediately note that \DUrole{raw-tex}{$(\ref{eq13})$} implies that the zero energy
state (the vacuum)  comes in degenerate pairs of the states having the same energy, where one member is
a boson and the other one is  a fermion.
This is shown in the  equations \DUrole{raw-tex}{$(\ref{eq15a})$} :

\begin{flalign}\label{eq15a}
 |0> = |E,\lambda>,\, E=0, \nonumber \\
 E=1/4Tr<0|\{Q_{\alpha},\bar{Q}_{\alpha}\}|0>=1/4|Q_{\alpha}|0>|^2 = 0, \,\, if \nonumber \\
  Q_{\alpha}|0>=0,\,\,\,\bar{Q}_{\alpha}|0>=|E,\lambda+1/2>,\,\,\bar{Q}_{\alpha}^i\bar{Q}_\beta^j|0>=|E,\lambda+1> \,\,\, etc
\end{flalign}

\DUrole{raw-tex}{\noindent}
This the degeneracy of the vacum  is destroyed,  if the invariance of the   vacum  is spontaneously broken, i.e
\DUrole{raw-tex}{$$ Q_{\alpha}|0>=|E,\lambda-1/2> \neq 0, \,\, E>0,\,\, \bar{Q}_{\alpha}|0>=|E^{\prime},\lambda+1/2>,\,\, E\neq E^{\prime} $$}
Here \DUrole{raw-tex}{$\lambda$} is a helicity of ground state. The \DUrole{raw-tex}{$N-$} supersymmetry considers  the vacum as the  multipartilce state
\DUrole{raw-tex}{$$\bar{Q}_1\bar{Q}_2\bar{Q}_3...\bar{Q}_N|0>=|E,\lambda+N/2>$$}.  Thus number of supersymmetries \DUrole{raw-tex}{$N$}
relates to the maximal spin of the particle \DUrole{raw-tex}{$S$}  in the multiplet as \DUrole{raw-tex}{$$ -S+N/2\leq S \Rightarrow N\leq 4S. $$}
Here  multiplet is assumed to be invariant under CPT transformation, i.e any state is doubly degenerated
\DUrole{raw-tex}{$|E,\pm\lamda(\pm 1/2;\pm 1; etc)>$}.
The total number of the possible states in
the \DUrole{raw-tex}{$N-$} supersymmetry theory is \DUrole{raw-tex}{$2^N=2^{N-1}\,\,bosons+2^{N-1}\,\,fermions$}
The quantum field theories with \DUrole{raw-tex}{$S>1$} are non-renormalizable. Hence I will restrict the consideration of SUSY to \DUrole{raw-tex}{$N =  1$}
which contains two types of the supermultiplets \DUrole{raw-tex}{$(\ref{eq15b})$}: chiral and vector multiplets

\begin{flalign}\label{eq15b}
 Scalar=\phi = |E,\lambda= 0>,\, Fermion=\xi=\bar{Q}|\phi> = |E,\lambda=1/2>,   \nonumber \\
 Fermion=\xi=|E,\lambda= 1/2>,\, Vector=A_{\mu}=\bar{Q}|\xi> = |E,\lambda=1>
\end{flalign}

\DUrole{raw-tex}{\noindent}
Thus in any supersymmetric theory,
every particle has a partner with the same mass but with a spin differing by \DUrole{raw-tex}{$1/2$}
(since \DUrole{raw-tex}{$Q$} is  a spinorial operator)
SUSY acts independently of any internal symmetry. In other words, the
generators of supersymmetry commute with all internal symmetry generators.
As a result, any particle and its superpartner have identical
internal quantum numbers such as electric charge, isospin, colour, etc.


\subsection{2.2~~~Superfields.%
  \label{superfields}%
}

A construction of SUSY invariant Lagrangiants requires to introduce chiral and vector superfields. Their
formalism  can  easily be used to write out the general SUSY invariant  Lagrangian with local gauge
interaction.
In general, the superfield is an anlytic function \DUrole{raw-tex}{$\mathcal{F}(z)$} defined in the superspace \DUrole{raw-tex}{$z$}. The function contains terms which
are proportional Grassmanians in some power of \DUrole{raw-tex}{$\theta,\bar{\theta}$} up to 4 at most: \DUrole{raw-tex}{$1,\theta, \bar{\theta},\theta\theta, \bar{\theta}\bar{\theta},
\theta\theta\bar{\theta}, etc$} . The most general view of the superfield is

\begin{flalign}\label{eq166}
\mathcal{F}(z)\equiv \mathcal{F}(x,\ \theta,\overline{\theta})\ =\ f(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}\overline{\chi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}N(x)+ \nonumber \\
\theta\sigma^{\mu}\overline{\theta}A_{\mu}(x)+\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}\theta\zeta(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
This is the most general form since \DUrole{raw-tex}{$\overline{\theta}\overline{\sigma}^{\mu}\theta=-\theta\sigma^{\mu}\overline{\theta}$}
and \DUrole{raw-tex}{$\theta\sigma^{\mu\nu}\theta=0=\overline{\theta}\overline{\sigma}^{\mu\nu}\overline{\theta}$}.
The \DUrole{raw-tex}{$\sqrt{2}$} coefficients have been
put in for convenience with respect to supersymmetry transformations.
One can consider the hermitian conjugate \DUrole{raw-tex}{$\mathcal{F}^{\dagger}(z)$} as
an independent superfield.
\DUrole{raw-tex}{$\mathcal{F}(z)$} doesn't need to have a well-defined parity. There are  several component fields of
the superfield \DUrole{raw-tex}{$\mathcal{F}(x,\ \theta,\overline{\theta})$}. They are
scalar fields \DUrole{raw-tex}{$f(x),\ M(x),\ N(x)$} and \DUrole{raw-tex}{$D(x)$}, one vector field \DUrole{raw-tex}{$A_{\mu}(x)$} plus two left handed Weyl spinor
fields \$xi\_\{A\}(x),zeta\_\{A\}(x)\$ and two right handed Weyl spinor fields
\DUrole{raw-tex}{$\overline{\chi}^{A}(x)$} and \DUrole{raw-tex}{$\overline{\lambda}^{\dot{A}}(x)$}.
All fields being complex, there are sixteen real bosonic and sixteen real fermionic component fields.
These are, of course, off-shell degrees of freedom since we have not imposed the equations of motion.
They  correspond (though not necessarily in a one to one way) to particles that neatly fall into complete supermultiplets.
The effect of the infinitesimal supersymmetry transformation \DUrole{raw-tex}{$\ref{eq10}$} on the components of \DUrole{raw-tex}{$\mathcal{F}$}
can be worked out \DUrole{raw-tex}{\cite{Salam:1974yz},\cite{Salam:1974jj},\cite{Wess:1974jb}}
from the requirement that \DUrole{raw-tex}{$\delta \mathcal{F}$} has the same form as  the superfield \DUrole{raw-tex}{$(\ref{eq166})$}. Thus
the variation of components obey \DUrole{raw-tex}{$(\ref{eq16})$}

\begin{flalign}\label{eq16}
\delta \mathcal{F}\equiv  \mathcal{F}(x^{\mu}-i\theta\sigma^{\mu}\overline{\epsilon}+
i\epsilon\sigma^{\mu}\overline{\theta},\ \theta+\epsilon,\overline{\theta}+
\overline{\epsilon})-\mathcal{F}(x,\ \theta,\overline{\theta})=i(\epsilon Q+\overline{\epsilon}\overline{Q})\mathcal{F},\nonumber \\
\delta f=\sqrt{2}\epsilon\xi+\sqrt{2}\overline{\epsilon}\overline{\chi}, \nonumber \\
\delta(\sqrt{2}\xi_{A})=2\epsilon_{A}M+(\sigma^{\mu}\overline{\epsilon})_{A}(-i\partial_{\mu}f+A_{\mu}), \nonumber \\
\delta(\sqrt{2}\overline{\chi}^{\dot{A}})=2\overline{\epsilon}^{A}N-(\overline{\sigma}^{\mu}\epsilon)^{A}(i\partial_{\mu}f+A_{\mu}), \nonumber \\
\displaystyle \delta M=\overline{\epsilon}\overline{\lambda}+\frac{i}{\sqrt{2}}\partial_{\mu}\xi\sigma^{\mu}\overline{\epsilon}, \nonumber \\
\displaystyle \delta N=\epsilon\zeta-\frac{i}{\sqrt{2}}\epsilon\sigma^{\mu}\partial_{\mu}\overline{\chi}, \nonumber \\
\delta \mathrm{A}_{\mu}\ =\ \epsilon\sigma_{\mu}\overline{\lambda}+\zeta\sigma_{\mu}\overline{\epsilon}-\frac{i}{\sqrt{2}}\epsilon\partial_{\mu}\xi+\frac{i}{\sqrt{2}}\partial_{\mu}\overline{\chi}\overline{\epsilon}, \nonumber  \\
+\sqrt{2}\epsilon\sigma_{\mu\nu}\partial^{\nu}\xi-\sqrt{2}\overline{\epsilon}\overline{\sigma}_{\mu\nu}\partial^{\nu}\overline{\chi},   \nonumber \\
\displaystyle \delta\overline{\lambda}^{A}=\overline{\epsilon}^{A}D-\frac{i}{2}\overline{\epsilon}^{A}\partial^{\mu}A_{\mu}-i(\overline{\sigma}^{\mu}\epsilon)^{A}\partial_{\mu}M+(\overline{\sigma}^{\mu\nu}\overline{\epsilon})^{A}\partial_{\mu}A_{\nu}, \nonumber \\
\displaystyle \delta\zeta_{A}=\epsilon_{A}D+\frac{i}{2}\epsilon_{A}\partial^{\mu}A_{\mu}-i(\sigma^{\mu}\overline{\epsilon})_{A}\partial_{\mu}N-(\sigma^{\mu\nu}\epsilon)_{A}\partial_{\mu}A_{\nu}, \nonumber \\
\delta D=i\partial_{\mu}(\zeta\sigma^{\mu}\overline{\epsilon}+\overline{\lambda}\overline{\sigma}^{\mu}\epsilon)
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$\delta D$} being a four divergence, with the usual assumption of discarding surface terms,
(i.e. coefficient of \DUrole{raw-tex}{$\theta\theta\overline{\theta}\overline{\theta}$} in a superfield)
in the Lagrangian density would yield a supersymmetric action.
Linear combinations of superfields are again superfields (since \DUrole{raw-tex}{$Q,\overline{Q}$} are linear differential operators),
i.e. superfields form linear representations of the supersymmetry algebra.
Products of superfields will be general superfields.
Some of the component fields (called auxiliary fields, for example \DUrole{raw-tex}{$M,N,D$}) do not contribute to an on-shell description,
and can be ruled out by impliying the equation of motion.
Superfield representations are highly reducible.
i.e. are physically redundant. It's possible to obtain irreducible representaion if someone introduce additional constraints
on  the supermultiplet.
Specific types of irreducible superfields are
chiral and vector superfields. However, products of irreducible superfields may or may not be irreducible superfields.


\subsubsection{2.2.1~~~Chiral superfields.%
  \label{chiral-superfields}%
}

I'm going to introduce chiral and vector supermultilet as building blocks of the SUSY invariant Lagrangian.
First, left and right chiral covariant derivatives \DUrole{raw-tex}{$\partial_A$} will be considered \DUrole{raw-tex}{\cite{Wess:1974jb}}. The spinorial derivative is not conserved
under the supersymmetric transformation
\DUrole{raw-tex}{$x^{\prime\mu}\equiv x^{\mu}-i\theta\sigma^{\mu}\overline{\epsilon}+i\epsilon\sigma^{\mu}\overline{\theta},\ \theta'\equiv\theta+\epsilon,\overline{\theta}'\equiv\overline{\theta}+\overline{\epsilon}$}
\DUrole{raw-tex}{$$ \partial_{A}=\frac{\partial\theta^{\prime B}}{\partial\theta^{A}}\frac{\partial}{\partial\theta^{\prime B}}+\frac{\partial x^{\prime\mu}}{\partial\theta^{A}}\frac{\partial}{\partial x^{\prime\mu}}
=\frac{\partial}{\partial\theta^{A}}-i(\sigma^{\mu}\overline{\epsilon})_{A}\frac{\partial}{\partial x^{\mu}} \ \ \
\frac{\partial}{\partial^{\prime}_A} \neq   \frac{\partial}{\partial_A}
$$}. Also \DUrole{raw-tex}{$\partial_A$} doesn't commute with the supersymmetry charges \DUrole{raw-tex}{$Q$}, \DUrole{raw-tex}{$ \bar{Q}$} in
contradiction to the momentum operator \DUrole{raw-tex}{$\partial_{\mu}=-iP_{\mu}$}. Thus the  left and right chiral covariant derivatives
defined as   \DUrole{raw-tex}{$(\ref{eq17})$}

\begin{flalign}\label{eq17}
 D_{A}\equiv\partial_{A}-i\sigma_{AB}^{\mu}\overline{\theta}^{B}\partial_{\mu}, \nonumber \\
 \overline{D}_{A}\equiv -\overline{\partial}_{A}+i\theta^{B}\sigma_{BA}^{\mu}\partial_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
anticommutes with  \DUrole{raw-tex}{$Q$}, \DUrole{raw-tex}{$\bar{Q}$} operators    as shown in \DUrole{raw-tex}{$(\ref{eq18})$}

\begin{flalign}\label{eq18}
\left\{ D_{A},\bar{Q}_B\right\} = \left\{\bar{D}_A,Q_B \right\}=0, \nonumber \\
\left\{ \bar{D}_{A},\bar{Q}_B\right\} = \left\{D_A,Q_B \right\}=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
Also, it follows from \DUrole{raw-tex}{$(\ref{eq18})$}  that \DUrole{raw-tex}{$D_A(\bar{D_A})$} is invariant under supersymmetric transformation \DUrole{raw-tex}{$\delta =
i(\epsilon Q + \bar{\epsilon}\bar{Q})$}

\begin{flalign}\label{eq19}
  D_A (x^{\prime},\theta^{\prime},\bar{\theta}^{\prime} )  = D_A (x,\theta,\bar{\theta} ), \nonumber \\
 \bar{D}_A (x^{\prime},\theta^{\prime},\bar{\theta}^{\prime} ) = \bar{D}_A (x,\theta,\bar{\theta} ), \nonumber \\
 \delta (\bar{D}_A \Phi(z)) = \delta (\bar{D}_A) \Phi(z) + \bar{D}_A  \delta (\Phi(z)) = \bar{D}_A  \delta (\Phi(z))
\end{flaign}

\DUrole{raw-tex}{\noindent}
If someone performs the shift \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\bar{\theta}$} or \DUrole{raw-tex}{$\overline{y}^{\mu}
\equiv x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$} in superspace \DUrole{raw-tex}{$z,$} then the
right (left) chiral covariant derivative \DUrole{raw-tex}{$\bar{D}_A$} (\DUrole{raw-tex}{$D_A$}) is zero for
any analytic function \DUrole{raw-tex}{$f(y,\theta)\,\,(f^{\star}(\bar{y},\bar{\theta}))$} in the point \DUrole{raw-tex}{$y(\overline{y})$}

\begin{flalign}\label{eq20}
\overline{D}_{A} y^{\mu}=0, D_{A}\overline{y}^{\mu}=0, \nonumber \\
\overline{D}_{A} f(y,\ \theta)= 0, D_{A}f^{\star}(\overline{y},\overline{\theta})=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
A multiplet \DUrole{raw-tex}{$\Phi(y,\theta)\,\,(\Phi^{+}(\bar{y},\bar{\theta}))$} is left (right) chiral superfield,
if it  obeys the equation \DUrole{raw-tex}{$(\ref{eq21})$}

\begin{flalign}\label{eq21}
 \bar{D}\Phi (y,\theta) =0, \nonumber \\
 D \Phi^{\dagger} (\bar{y},\bar{\theta}) =0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The decomposition of these superfields in terms of their component fields
can be given as

\begin{flalign}\label{eq21}
 \Phi(y,\ \theta)=\phi(y)+\sqrt{2}\theta\xi(y)+\theta\theta F(y),\nonumber \\
 \Phi^{\dagger}(\overline{y}_{)}\overline{\theta})=
 \phi^{\star}(\overline{y})+\sqrt{2}\overline{\theta}\overline{\xi}(\overline{y})+
 \overline{\theta}\overline{\theta}F^{\star}(\overline{y})
\end{flalign}

\DUrole{raw-tex}{\noindent}
Taylor expansion of \DUrole{raw-tex}{(\ref{eq21})}  using substitutions \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\overline{\theta}$}, and
\DUrole{raw-tex}{$\overline{y}^{\mu}=x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$} gives us

\begin{flalign}\label{eq22}
\Phi(y,\ \theta)\ =\ \phi(x)-i\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\phi(x)-
\frac{1}{4}\theta\theta\overline{\theta}\overline{\theta}\partial^{\mu}\partial_{\mu}\phi(x)+\sqrt{2}\theta\xi(x), \nonumber \\
 +\displaystyle \frac{i}{\sqrt{2}}\theta\theta\partial_{\mu}\xi\sigma^{\mu}\overline{\theta}+\theta\theta F(x),  \nonumber \\
 \Phi^{\dagger}(\overline{y},\overline{\theta})\ =\ \phi^{\star}(x)+i\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\phi^{\star}(x)-\frac{1}{4}\theta\theta\overline{\theta}
 \overline{\theta}\partial^{\mu}\partial_{\mu}\phi^{\star}(x)+\sqrt{2}\overline{\theta}\overline{\xi}(x) \nonumber \\
 -\displaystyle \frac{i}{\sqrt{2}}\overline{\theta}\overline{\theta}\theta\sigma^{\mu}\partial_{\mu}\overline{\xi}(x)+\overline{\theta}\overline{\theta}F^{\star}(x)
\end{flalign}

\DUrole{raw-tex}{\noindent}
Comparing \DUrole{raw-tex}{$(\ref{eq22})$} with \DUrole{raw-tex}{$(\ref{eq16})$}, the supersymmetric transformations of the chiral components can be expressed as

\begin{flalign}\label{eq23}
\delta\phi=\sqrt{2}\epsilon\xi, \nonumber  \\
\delta\xi_{A}=\sqrt{2}\epsilon_{A}F-\sqrt{2}i(\sigma^{\mu}\overline{\epsilon})_{A}\partial_{\mu}\phi, \nonumber \\
\delta F=i\partial_{\mu}(\sqrt{2}\xi\sigma^{\mu}\overline{\epsilon})
\end{flalign}

\DUrole{raw-tex}{\noindent}
Component fields in \DUrole{raw-tex}{$\Phi\dagger$} obey the corresponding hermitian conjugate transformations.
The \DUrole{raw-tex}{$F$}-component of \DUrole{raw-tex}{$\Phi$} transforms into itself plus a spacetime derivative.
Hence, such a term in the Lagrangian density, called an F-term,
leads to a supersymmetry invariant action when surface terms can be discarded.
Products of chiral superfields \DUrole{raw-tex}{$\Phi_{1}\Phi_{2}\cdots\Phi_{l}$} or \DUrole{raw-tex}{$\Phi_{1}^{\dagger}\Phi_{2}^{\dagger}
\cdots\Phi_{l}^{\dagger}$} are also chiral superfields themselves as shown in \DUrole{raw-tex}{$(\ref{eq24})$}

\begin{flalign}\label{eq24}
\Phi_{i}\Phi_{j}=\phi_{i}\phi_{j}+\sqrt{2}\theta(\xi_{i}\phi_{j}+\phi_{i}\xi_{j})+\theta\theta(\phi_{i}F_{j}+\phi_{j}F_{i}-\xi_{i}\xi_{j}), \nonumber \\
\Phi_{i}\Phi_{j}\Phi_{k}\ =\ \phi_{i}\phi_{j}\phi_{k}+\sqrt{2}\theta(\xi_{i}\phi_{j}\phi_{k}+\xi_{j}\phi_{k}\phi_{i}+\xi_{k}\phi_{i}\phi_{j}) \nonumber \\
+\theta\theta(F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j}-\xi_{i}\xi_{j}\phi_{k}-\xi_{j}\xi_{k}\phi_{i}-\xi_{k}\xi_{i}\phi_{j}), \nonumber \\
\bar{D}(\Phi_{i}\Phi_{j})= \bar{D} (\Phi_{i}\Phi_{j}\Phi_{k}) = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The important point to note is that in the Lagrangian density
the \DUrole{raw-tex}{$F$}-term in any polynomial function of chiral superfields would yield a supersymmetric action
\DUrole{raw-tex}{$S=\displaystyle \int d^4xd^2\theta\Phi_{1}(z)\cdots\Phi_{n}(z),$} where  the Lagrangian density
\DUrole{raw-tex}{$\mathcal{L}$} is \DUrole{raw-tex}{$\Phi_{1}(z)\cdots\Phi_{n}(z) |_{\theta\theta}$} (\DUrole{raw-tex}{$ |_{\theta\theta}$} denotes \DUrole{raw-tex}{$F$}-term).
Such is not the case for the product of a chiral superfield with
its hermitian conjugate \DUrole{raw-tex}{$\Phi_{i}^{\dagger}\Phi_{j}$}. The product is a vector superfield which \DUrole{raw-tex}{$D$}-component,
coefficient of \DUrole{raw-tex}{$\theta\theta\overline{\theta}\overline{\theta}$}
(\DUrole{raw-tex}{$ |_{\theta\theta\overline{\theta}\overline{\theta}}$}), appears in a Lagrangian density yeilding a supersymmetric
action \DUrole{raw-tex}{$S=\int d^{4}xd^2\theta d^2\bar{\theta}\Phi_{i}^{\dagger}\Phi_{j}=\int d^4 x \Phi_{i}^{\dagger}\Phi_{j}
|_{\theta\theta\overline{\theta}\overline{\theta}}$}.


\subsubsection{2.2.2~~~Vector superfields.%
  \label{vector-superfields}%
}

A vector superfield  requires reality \DUrole{raw-tex}{$$ V = V^{\dagger}. $$}
Having in mind the general form of superfield \DUrole{raw-tex}{$(\ref{eq16})$}, the vector superfield \DUrole{raw-tex}{$V(x,\theta\bar{\theta})$}
is expressed as the following

\begin{flalign}\label{eq25}
 V(x,\ \theta,\overline{\theta})\ \sim\ C(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}
 \overline{\xi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}M^{\star}(x)+
 \theta\sigma^{\mu}\overline{\theta}A_{\mu}(x) \nonumber \\
 +\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}
  \theta\lambda(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x),
\end{flalign}

\DUrole{raw-tex}{\noindent} where \DUrole{raw-tex}{$C(x),\ A_{\mu}(x)$} and \DUrole{raw-tex}{$D(x)$} are real fields, \DUrole{raw-tex}{$M(x)$} is a complex scalar field
and \DUrole{raw-tex}{$\xi(x),\ \lambda(x)$} are complex two component spinor fields. Vector superfields can be constructed
from a chiral superfield \DUrole{raw-tex}{$\Phi$} in the way like \DUrole{raw-tex}{$$ V\sim \Phi + \Phi^{\dagger}, \ \  \sim \Phi^{\dagger}\Phi $$}
Using \DUrole{raw-tex}{$(\ref{eq22})$}, and replacing \DUrole{raw-tex}{$\xi(x)$} by \DUrole{raw-tex}{$\chi(x)$}, we can write

 \begin{flalign}\label{eq26}
  \Phi + \Phi^{\dagger} = 2 \Re \phi(x)+\sqrt{2}\theta\chi(x)+ \sqrt{2}\overline{\theta}
 \overline{\chi}(x)+\theta\theta F(x)+\overline{\theta}\overline{\theta}F^{\star}(x) \nonumber \\
+2\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\Im\phi(x)-\frac{i}{\sqrt{2}}
\theta\theta\overline{\theta}\overline{\sigma}^{\mu}\partial_{\mu}\chi(x)-\frac{i}{\sqrt{2}}
\overline{\theta}\overline{\theta}\theta\sigma^{\mu}\partial_{\mu}\overline{\chi}(x) \nonumber \\
- \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\partial^{\mu}\partial_{\mu}\Re\phi(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The definition \DUrole{raw-tex}{$(\ref{eq25})$} can be  rewritten  with addtitional substitutions \DUrole{raw-tex}{$\lambda-i\sigma^{\mu}\partial_{\mu}
\overline{\xi}/\sqrt{2}$} for
\DUrole{raw-tex}{$\lambda,\ \overline{\lambda}-i\overline{\sigma}^{\mu}\partial_{\mu}\xi/\sqrt{2}$} for \DUrole{raw-tex}{$\overline{\lambda}$} and
\DUrole{raw-tex}{$D-\displaystyle \frac{1}{2}\partial^{\mu}\partial_{\mu}C$} for \DUrole{raw-tex}{$D$}. Thus a more general vector superfield  defined as

\begin{flalign}\label{eq27}
V(z)\ =\ C(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}\overline{\xi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}M^{\star}(x)+\theta\sigma^{\mu}\overline{\theta}\mathrm{A}_{\mu}(x) \nonumber \\
+\theta\theta\overline{\theta}\{\overline{\lambda}(x)-\frac{i}{\sqrt{2}}\overline{\sigma}^{\mu}\partial_{\mu}\xi(x)\}+\overline{\theta}\overline{\theta}\theta\{\lambda(x)-\frac{i}{\sqrt{2}}\sigma^{\mu}\partial_{\mu}\overline{\xi}(x)\} \nonumber \\
+ \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(x)-\frac{1}{2}\partial^{\mu}\partial_{\mu}C(x)\}
\end{flalign}

\DUrole{raw-tex}{\noindent}
can be \emph{supergauge transformed} with changed field components \DUrole{raw-tex}{$ C\rightarrow C+2\Re e\phi,\ \xi\rightarrow\xi+\chi,\ M\rightarrow M+F,\
A_{\mu}\rightarrow A_{\mu}-2\partial_{\mu}\Im m\phi,\ \lambda\rightarrow\lambda,\ D\rightarrow D$}. There is a particular gauge,
\emph{Wess-Zummino gauge} \DUrole{raw-tex}{\cite{WZ}}, where the vector  superfield reduces to the form \DUrole{raw-tex}{$$V(z)=(0,0,0,\ A_{\mu},\ \lambda,\ D). $$}
The gauge is very convienient to write out supersymmetric Lagrangians. One interesting property of the Wess-Zumino
gauge is the ease with which powers of \DUrole{raw-tex}{$V(z)$} can be calculated \DUrole{raw-tex}{$(\ref{eq28})$}.

\begin{flalign}\label{eq28}
V(z)=\displaystyle \theta\sigma^{\mu}\overline{\theta}A_{\mu}(x)+\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}\theta\lambda(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x) \nonumber \\
V^2(z) = \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}A^{\mu}A_{\mu},\nonumber \\
V^{n}(z)=0\forall n\geq 3
 \end{flalign}

The last point concerning the vector superfields is the expression of the field  strength which remains invariant under
supergauge transformation. Someone can construct left and right chiral field-strength superfields

\begin{flalign}\label{eq29}
 W_{A}=-\displaystyle \frac{1}{4}\bar{D}^{B}\bar{D}_{B}D_{A}V(x,\theta,\bar{\theta}) = -\frac{1}{4}\bar{D}\bar{D}D_{A}V, \nonumber \\
 \overline{W}_{A}=-\frac{1}{4}DD\overline{D}_{A}V
 \end{flalign}

\DUrole{raw-tex}{\noindent}
which, in accordance with \DUrole{raw-tex}{$(\ref{eq18})$},  obeys the following equations \DUrole{raw-tex}{$(\ref{eq30})$} in

\begin{flalign}\label{eq30}
\overline{D}_{A}W_{A}=0,\ \ D_{A}\overline{W}_{A}=0 \nonumber \\
 V^{'} = V+\Phi + \Phi^{\dagger}, \ \ W_A^{'} = \frac{1}{4}\bar{D}\bar{D}D_AV^{'}=
 \frac{1}{4}(\bar{D}(\{\bar{D},D_A\}-D_A\bar{D})(V+\Phi + \Phi^{\dagger}) = \nonumber \\
 \frac{1}{4}\bar{D}(\bar{D}D_AV+\bar{D}\Phi + D_A\Phi^{\dagger}) =  \frac{1}{4}\bar{D}\bar{D}D_AV = W_A
 \end{flalign}

\DUrole{raw-tex}{\noindent}
It is convenient to use \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\bar{\theta}$} or \DUrole{raw-tex}{$\overline{y}^{\mu}=x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$}
shifts to write out the vector field and the field strength  in the simplest way

\begin{flalign}\label{eq31}
 V(y,\displaystyle \ \theta,\overline{\theta})=\theta\sigma^{\mu}\overline{\theta}A_{\mu}(y)+\theta\theta\overline{\theta}\overline{\lambda}(y)+\overline{\theta}\overline{\theta}\theta\lambda(y)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(y)+i\partial_{\mu}^{(y)}A^{\mu}(y)\}\nonumber \\
 V(\displaystyle \overline{y},\ \theta,\overline{\theta})=\theta\sigma^{\mu}\overline{\theta}A_{\mu}(\overline{y})+\theta\theta\overline{\theta}\overline{\lambda}(\overline{y})+\overline{\theta}\overline{\theta}\theta\lambda(\overline{y})+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(\overline{y})-i\partial_{\mu}^{(\overline{y})}A^{\mu}(\overline{y})\}\nonumber \\
 \overline{D}(y) = -\overline{\partial_A},\ \ \  D(y)=\partial_A - 2i\sigma^{\mu}_{AB}\overline{\theta}^B\partial_{\mu}, \ \ \ D(\overline{y}) = \partial_A\nonumber \\
 W_A = -\frac{1}{4}\bar{D}(y)\bar{D}(y)D_{A}(y)V(y,\theta,\bar{\theta})=
 \lambda_{A}(y) +D(y)\theta_{A}-(\sigma^{\mu\nu}\theta)_{A}F_{\mu\nu}(y)+i\theta\theta\sigma_{A\dot{B}}^{\mu}\partial_{\mu}^{(y)}\overline{\lambda}^{B}(y), \nonumber \\
 \overline{W}_A = \overline{\lambda}_{\dot{A}}(\overline{y})+D(\overline{y})\overline{\theta}_{\dot{A}}-(\overline{\sigma}^{\mu\nu}\overline{\theta})_{A}F_{\mu\nu}(\overline{y})-i\overline{\theta}\overline{\theta}\{\partial_{\mu}^{(\overline{y})}\lambda(\overline{y})\sigma^{\mu}\}_{A}
 \end{flalign}

\DUrole{raw-tex}{\noindent}
where  \DUrole{raw-tex}{$$F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$}
The kinetic term \DUrole{raw-tex}{$1/4 |W_AW^A + \overline{W}_A\overline{W}^A|_{\theta\theta,\bar{\theta}\bar{\theta}}$} in the SUSY
Lagrangian takes the familar form \DUrole{raw-tex}{$(\ref{eq32})$}

\begin{flalign}\label{eq32}
 W^{A}W_{A}\ =\ \lambda(y)\lambda(y)+2\theta\{D(y)\lambda(y)+\sigma^{\mu\nu}\lambda(y)F_{\mu\nu}(y)\}+ \nonumber \\
 \theta\theta\{D^{2}(y)+2i\lambda(y)\sigma^{\mu}\partial_{\mu}^{(y)}\overline{\lambda}(y)-\frac{1}{2}F_{\mu\nu}(y)F^{\mu\nu}(y) -  \nonumber \\
 \displaystyle \frac{i}{2}\tilde{F}_{\mu\nu}(y)F^{\mu\nu}(y), \nonumber \\
 \overline{W}_{A}\overline{W}^{A}\ =\ \overline{\lambda}(\overline{y})\overline{\lambda}(\overline{y})+\{2D(\overline{y})\overline{\lambda}(\overline{y})
 +2\overline{\lambda}(\overline{y})\overline{\sigma}^{\mu\nu}F_{\mu\nu}(\overline{y})\}\overline{\theta} + \nonumber \\
 \overline{\theta}\overline{\theta}\{D^{2}(\overline{y})-2i\partial_{\mu}^{(\overline{y})}\lambda(\overline{y})\sigma^{\mu}\overline{\lambda}(y)-\frac{1}{2}F_{\mu\nu}(\overline{y})F^{\mu\nu}(\overline{y}) + \nonumber \\
 \displaystyle \frac{i}{2}\tilde{F}_{\mu\nu}(\overline{y})F^{\mu\nu}(\overline{y}),\nonumber \\
 \displaystyle \frac{1}{4}[W^{A}W_{A}+\overline{W}_{A}\overline{W}^{A}]_{\theta\theta,\overline\theta\overline\theta}=\frac{1}{2}D^{2}(x)-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)+i\lambda(x)\sigma^{\mu}[\partial_{\mu}]\overline{\lambda}(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
In the last line of \DUrole{raw-tex}{$(\ref{eq32})$}, the back substitution \DUrole{raw-tex}{$y\rightarrow x$} was done in \DUrole{raw-tex}{$F-$} term because
the 4-dimensional intergral
is invariant under all SUSY transformations \DUrole{raw-tex}{$\int d^4x W^{A}W_{A}|_{\theta\theta}  \equiv \int d^4y W^{A}W_{A}|_{\theta\theta}$}


\subsubsection{2.2.3~~~R-parity of the chiral and vector superfields%
  \label{r-parity-of-the-chiral-and-vector-superfields}%
}

There is \DUrole{raw-tex}{$R-$} symmetry as a global \DUrole{raw-tex}{$U(1)$} transformation in SUSY which leaves Super-Poincare Lie Algebra \DUrole{raw-tex}{$(\ref{eq11}),(\ref{eq12}),(\ref{eq13}),(\ref{eq14})$}
unchanged.

 \begin{flalign}\label{eq33}
 Q_{A}\rightarrow e^{i\varphi R}Q_{A}e^{-i\varphi R}=e^{-i\varphi}Q_{A}, \nonumber\\
\overline{Q}_{A}\rightarrow e^{i\varphi R}\overline{Q}_{A}e^{-i\varphi R}=e^{i\varphi}\overline{Q}_{A}, \nonumber\\
 \theta\rightarrow e^{i\varphi}\theta,\ \ \ \overline{\theta} \rightarrow e^{-i\varphi}\overline{\theta}, \nonumber \\
[Q_A,R]=Q_A , \nonumber \\
[\overline{Q}_A,R]=-\overline{Q}_A
 \end{flalign}

\DUrole{raw-tex}{\noindent}
This \DUrole{raw-tex}{$(\ref{eq33})$} leads that  new quantum numbers, \DUrole{raw-tex}{$R-$} charges, can be assigned to
\DUrole{raw-tex}{$\theta,\bar{\theta},Q,\bar{Q},\Phi,\Phi^{\dagger}$} ,
which are \DUrole{raw-tex}{$1,-1,-1,1,R_{\Phi},-R_{\Phi}$} respectively, where  \DUrole{raw-tex}{$R_{\Phi}$} are the \DUrole{raw-tex}{$R-$} charges for the left and
rifht chiral superfields  derived from their \DUrole{raw-tex}{$R-$} symmetry properties

\begin{flalign}\label{eq34}
\Phi^{\prime}(x,e^{i\varphi}\theta,e^{-i\varphi}\overline{\theta})\rightarrow e^{i\varphi R_{\Phi}}\Phi(x,\theta,\overline{\theta}), \nonumber\\
\Phi^{\prime\dagger}(x,e^{i\varphi}\theta,e^{-i\varphi}\overline{\theta})\rightarrow e^{-i\varphi R_{\Phi}}\Phi^{\dagger}(x,\theta,\overline{\theta}), \nonumber\\
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The vector field has zero \DUrole{raw-tex}{$R-$} charge because of its reality \DUrole{raw-tex}{$V(x)=V^{\dagger}(x)$}.
Someone also can write out \DUrole{raw-tex}{$R-$} charges of the chiral and vector components in \emph{Wees-Zummino gauge} using \DUrole{raw-tex}{$(\ref{eq34})$}
and the previous statement about \DUrole{raw-tex}{$V(x)$}

\begin{flalign}\label{eq35}
R(\phi)=R_{\Phi}, \nonumber \\
R(\xi)=-R(\overline{\xi})=R_{\Phi}-1, \nonumber \\
R(F)=R_{\Phi}-2, \nonumber \\
R(A_{\mu})=0, \nonumber \\
R(\lambda)=-R(\overline{\lambda})=1, \nonumber \\
R(D)=0
 \end{flalign}

\DUrole{raw-tex}{\noindent}

The supersymmetric Lagrangian with gauge interaction has a general form \DUrole{raw-tex}{$\ref{eq36}$}

\begin{flalign}\label{eq36}
 \mathcal{L} = \frac{1}{4}(\int d^2\theta W_AW^A + \int d^2\overline{\theta}\overline{W_A}\overline{W^A})  + \nonumber \\
 \int d^2\theta d^2\overline{\theta} \Phi^{\dagger}e^{V}\Phi + \int d^2\theta W(\Phi) + \int d^2\theta W^{*}(\Phi^{\dagger})
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The kinetic \DUrole{raw-tex}{$W_AW^A|_{\theta\theta}$} and gauge interaction \DUrole{raw-tex}{$\Phi^{\dagger}e^V\Phi|_{\theta\theta\overline{\theta}\overline{\theta}}$}
terms are \DUrole{raw-tex}{$R-$} invariant, ie they have total zero \DUrole{raw-tex}{$R-$} charge.  But the superpotentional \DUrole{raw-tex}{$W(\Phi)$}, some analytic function of  having 3 \textsuperscript{rd} power of \DUrole{raw-tex}{$\Phi(x)$} at most.
and describing the self-interactions of superfields, can preserve or not preserve \DUrole{raw-tex}{$R-$} invariance. The \DUrole{raw-tex}{$R-$} nonsymmetric
terms can be presented in the part of the Lagrangian correspoding to  Yukawa interaction  as well in soft supersymmetry  breaking
terms. The Yukawa  \DUrole{raw-tex}{$R_p-$} violating  terms are forbidden in SUSY generalization of the Standard Model because of their non-Lorentz
invariance. In the time, the soft supersimmetry breaking part contains Lorentz and gauge invariant mass terms of gaugino,
Majorana spinor \DUrole{raw-tex}{$\lambda,\overline{\lambda}$} component of the vector superfield \DUrole{raw-tex}{$V(x)$}
\DUrole{raw-tex}{$$ M\cdot(\overline{\lambda}\overline{\lambda} + \lambda\lambda), $$} which has \DUrole{raw-tex}{$R\neq 0.$}
Thus \DUrole{raw-tex}{$U(1)_R$} invariance  has to be too restrictive and \DUrole{raw-tex}{$U(1)_R$} transformaton has to be abandoned as symmetry but
its discrete subgroup \DUrole{raw-tex}{$Z_2$} which has only two possible group elements \DUrole{raw-tex}{$\varphi=\pm \pi$}, conserves
soft supersymmetric breaking terms \DUrole{raw-tex}{$$ Z_2: \,\,\, M\lambda^{\prime}\lambda^{\prime}=Me^{i2\pi}\lamda\lambda=M\lambda\lambda.$$}
The value of \DUrole{raw-tex}{$e^{i\pi R_{\Phi}}=(-1)^{R_{\Phi}}$} is called a matter parity \DUrole{raw-tex}{$M_p$} of the superfield, while the corresponded
value for the component field is called  \DUrole{raw-tex}{$R_p-$} parity of the that field. In accordance to \DUrole{raw-tex}{$(\ref{eq35})$} and
\DUrole{raw-tex}{$Z_2$} symmetry,
the values of \DUrole{raw-tex}{$R_{\Phi}$} for field components can be \DUrole{raw-tex}{$\pm 1`$ or `$0$}. The two options define two categories of the chiral superfields:
matterlike or quantalike. In both cases, component fields corresponded to \emph{Standard Model} (\emph{SM})
representations have positive values of
\DUrole{raw-tex}{$R_p$} or equally  the component field has \DUrole{raw-tex}{$R_{\Phi}=0$}. Such supermultiplets, which contain spin 1/2 \emph{SM}
particles, as quarks or leptons, but scalar superpartners, are matterlike.  Oppositely, the quantalike multiples consist of
bosonic \emph{SM} fields and fermionice sparticles. This property of \DUrole{raw-tex}{$R_{\Phi}$} can be ensured by the relation \DUrole{raw-tex}{$$R_{\Phi}=3(B-L),$$}
where \DUrole{raw-tex}{$B$} is a baryon number and \DUrole{raw-tex}{$L-$} lepton number associated with the superfield. Then \DUrole{raw-tex}{$M_p$} parity
is \DUrole{raw-tex}{$(-1)^{3(B-L)}$} while
\DUrole{raw-tex}{$R_p-$} parity is expressed as \DUrole{raw-tex}{$(-1)^{3(B-L)+2S},$} with \DUrole{raw-tex}{$S$} being the spin.

\begin{flalign}\label{eq366}
M_p=(-1)^{3(B-L)},\,\,\,\, R_p=(-1)^{3(B-L)+2S}
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$R_p-$} parity of a state containing several particles and spaiticles is the product of the individual parities.Thus
for the particles and sparticles under discussion,  \DUrole{raw-tex}{$R_{p}$} conservation in any interaction vertex
means that sparticle production processes must produce them in even numbers (usually a pair) and
every sparticle other than the \emph{Lightest Supersymmetric Particle} (LSP) will eventually decay into particles plus
an odd number of LSPs (usually one). I will consider  \emph{Minimal Supersymmetric Standard Model}  (MSSM) as   a such supersymmetry theory
with exact \DUrole{raw-tex}{$R-$} parity conservation.

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.185\DUtablewidth}|p{0.269\DUtablewidth}|p{0.058\DUtablewidth}|p{0.111\DUtablewidth}|p{0.164\DUtablewidth}|p{0.174\DUtablewidth}|}
\caption{\DUrole{raw-tex}{$R_p$} for particles and sparticles}\\
\hline
\textbf{%
Supermultiplet
} & \textbf{%
Name
} & \textbf{%
Spin
} & \textbf{%
\DUrole{raw-tex}{$R_p$}
} & \textbf{%
\DUrole{raw-tex}{$R_{\Phi}$}
} & \textbf{%
Superfild type
} \\
\hline
\endfirsthead
\caption[]{\DUrole{raw-tex}{$R_p$} for particles and sparticles (... continued)}\\
\hline
\textbf{%
Supermultiplet
} & \textbf{%
Name
} & \textbf{%
Spin
} & \textbf{%
\DUrole{raw-tex}{$R_p$}
} & \textbf{%
\DUrole{raw-tex}{$R_{\Phi}$}
} & \textbf{%
Superfild type
} \\
\hline
\endhead
\multicolumn{6}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

Particle

Sparticle
 & 
Quark,q

Squark, \DUrole{raw-tex}{$\tilde{q}$}
 & 
1/2

0
 & 
+1

-1
 & 
1

1
 & 
Chiral,

matterlike
 \\
\hline

Particle

Sparticle
 & 
Lepton, l

Slepton, \DUrole{raw-tex}{$\tilde{l}$}
 & 
1/2

0
 & 
+1

-1
 & 
1

1
 & 
Chiral,

matterlike
 \\
\hline

Particle

Sparticle
 & 
Higgs,H

Higgsino, \DUrole{raw-tex}{$\tilde{H}$}
 & 
0

1/2
 & 
+1

-1
 & 
0

0
 & 
Chiral,

quantalike
 \\
\hline

Particle

Sparticle
 & 
Gauge boson,g

Gaugino, \DUrole{raw-tex}{$\tilde{g}$}
 & 
1

1/2
 & 
+1

-1
 & 
0

0
 & 
Vector
 \\
\hline
\end{longtable}


\subsection{2.3~~~Construction of SUSY Lagrangians%
  \label{construction-of-susy-lagrangians}%
}

The general SUSY  Lagrangian \DUrole{raw-tex}{$(\ref{eq37})$} consists of the kinetic term, gauge interaction and   superpotential as it follows
from \DUrole{raw-tex}{$(\ref{eq36})$}.

\begin{flalign}\label{eq37}
 \mathcal{L} = \frac{1}{4} (W_AW^A|_{\theta\theta} + \overline{W_A}\overline{W^A}|_{\overline{\theta}\overline{\theta}}) + \nonumber \\
  \Phi^{\dagger}e^{V}\Phi|_{\theta\theta\overline{\theta}\overline{\theta}} + W(\Phi)|_{\theta\theta}
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential \DUrole{raw-tex}{$W(\Phi)$} is a polinomial  of chiral superfields which Taylor expansion looks like

\begin{flalign}\label{eq38}
 W(\Phi_{i})\ =\ W(\phi_{i}+\sqrt{2}\theta\xi_{i}+\theta\theta F) \nonumber \\
 = \displaystyle W(\phi_{i})+\frac{\partial W}{\partial \phi_{i}}\sqrt{2}\theta\xi_{i}+\theta\theta(  \nonumber \\
 \frac{\partial W}{\partial \phi_{i}}F_{i}-\frac{1}{2}\frac{\partial^{2}W}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}).
\end{flalign}

\DUrole{raw-tex}{\noindent}
In particular, the SUSY generalization of QED, with two chiral supermultiplets introduced to have left- and right-handed
fermions  and with  the superpotential  of the second order \DUrole{raw-tex}{$W(\Phi_{-},\Phi_{+})=m\Phi_{-}\Phi_{+}$} is

 \begin{flalign}\label{eq39}
 \mathcal{L}_{SUSYQED}\ =\ \frac{1}{4}(W^{\alpha}W_{\alpha}|_{\theta\theta}+W^{\alpha}W_{\alpha}|_{\overline{\theta}\overline{\theta}})\nonumber \\
+ \displaystyle (\Phi_{+}^{+}e^{gV}\Phi_{+}+\Phi_{-}^{+}e^{-gV}\Phi_{-})|_{\theta\theta\overline{\theta}\overline{\theta}} \nonumber \\
 +\  m(\Phi_{+}\Phi_{-}|_{\theta\theta} + \Phi_{+}^{+}\Phi_{-}^{+}|_{\overline{\theta}\overline{\theta}})
 \end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can get the expression of SUSY Lagrangian  explicitly in terms of component fields.
The  simplest SUSY  model proposed by  Wess and Zumino  \DUrole{raw-tex}{\cite{Wess:1974jb},\cite{Wess:1974tw}}
has the  Lagrangian of only one
left chiral superfield   without  gauge  interaction.
The common expression of \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$}
\DUrole{raw-tex}{$$ \mathcal{L}_{SUSY}\ =   \Phi^{\dagger}_{i}\Phi_{j}|_{\overline{\theta\theta}\theta\theta} +W(\Phi_{i}\Phi_{j})|_{\theta\theta} $$}

depends on the  kinetic and interaction terms given by \DUrole{raw-tex}{$(\ref{eq40})$}.and \DUrole{raw-tex}{$(\ref{eq41})$}.

\begin{flalign}\label{eq40}
\Phi_{i}^{\dagger}\Phi_{j}\ =\ \phi_{\mathrm{i}}^{\star}\phi_{j}+\sqrt{2}\theta\xi_{j}\phi_{i}^{\star}+\sqrt{2}\overline{\theta}\overline{\xi}_{i}\phi_{j}+\theta\theta\phi_{i}^{\star}F_{j}+\overline{\theta}\overline{\theta}F_{i}^{\star}\phi_{j}+2\overline{\theta}\overline{\xi}_{i}\theta\xi_{j} \nonumber \\
 +\sqrt{2}\theta\theta\overline{\theta}_{\dot{A}}(i\overline{\sigma}^{\mu\dot{A}B}\xi_{jB}[\partial_{\mu}]\phi_i^{\star}+\overline{\xi_i^{\dot{A}}}F_j)\nonumber \\
 -2i\theta\sigma^{\mu}\overline{\theta}\phi_{i}^{\star}[\partial_{\mu}]\phi_{j} \nonumber \\
 +\sqrt{2}\overline{\theta}\overline{\theta}\theta^{A}(i\sigma_{A\dot{B}}^{\mu}\overline{\xi}_{i}^{B}[\partial_{\mu}]\phi_{j}+\xi_{jA}F_{i}^{\star}) \nonumber \\
  + \theta\theta\overline{\theta}\overline{\theta}(F_{i}^{\star}F_{j}+\frac{1}{2}\partial_{\mu}\phi_{i}^{\star}[\partial^{\mu}]\phi_{j}-\frac{1}{2}\phi_{i}^{\star}[\partial_{\mu}]\partial^{\mu}\phi_{j}+i\xi_{j}\sigma^{\mu}[\partial_{\mu}]\overline{\xi}_{i}), \nonumber  \\
\Phi_{i}^{\dagger}\Phi_{j}|_{\overline{\theta\theta}\theta\theta} = F_{i}^{\star}F_{j}+\frac{1}{2}\partial_{\mu}\phi_{i}^{\star}[\partial^{\mu}]\phi_{j}-\frac{1}{2}\phi_{i}^{\star}[\partial_{\mu}]\partial^{\mu}\phi_{j}+i\xi_{j}\sigma^{\mu}[\partial_{\mu}]\overline{\xi}_{i},
\end{flalign}

\begin{flalign}\label{eq41}
 W(\Phi) = \lambda_{i}\Phi_{i}+\frac{1}{2}m_{ij}\Phi_{i}\Phi_{j}+\frac{1}{3}g_{ijk}\Phi_{i}\Phi_{j}\Phi_{k}, \nonumber \\
 W(\Phi)|_{\theta\theta} = \lambda_{i}F_{i} + \frac{1}{2}m_{ij}(\phi_{i}F_{j}+\phi_{j}F_{i}-\xi_{i}\xi_{j}) \nonumber \\
 +  \frac{1}{3} g_{ijk}  (F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j}-\xi_{i}\xi_{j}\phi_{k}-\xi_{j}\xi_{k}\phi_{i}-\xi_{k}\xi_{i}\phi_{j})
\end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential  is a polinomial of the chiral superfield \DUrole{raw-tex}{$\Phi_i(z)$} of third order \DUrole{raw-tex}{$(\ref{eq41})$} at most that make SUSY to be  regularized.
The auxiliary fields \DUrole{raw-tex}{$F_{i}$} can ruled be out from \DUrole{raw-tex}{$(\ref{eq40})$} by the equtation of motion \DUrole{raw-tex}{$(\ref{eq42})$}

\begin{flalign}\label{eq42}
\frac{\partial \mathcal{L}}{\partial F_{k}^{*}} = F_{k}+\lambda_{k}^{*}+m_{ik}^{*}\phi_{i}^{*}+y_{ijk}^{*}\phi_{i}^{*}\phi_{j}^{*}=0, \nonumber \\
 \frac{\partial \mathcal{L}}{\partial F_{k}} = F_{k}^{*}+\lambda_{k}+m_{ik}\phi_{i}+y_{ijk}\phi_{i}\phi_{j}=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential  \DUrole{raw-tex}{$W(\Phi)$} in Wess-Zumino's model,  \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$}, is supposed
to have  the form \DUrole{raw-tex}{$$W(\Phi)=\frac{1}{2}m\Phi\Phi+\frac{g}{3}\Phi\Phi\Phi+h.c.$$} .
Using Taylor expansion \DUrole{raw-tex}{$(\ref{eq38})$}  of the superpotential \DUrole{raw-tex}{$W(\Phi)$}, the equation of
motion \DUrole{raw-tex}{$(\ref{eq42})$} is transformed to the simplest form as shown in  \DUrole{raw-tex}{$(\ref{eq43})$}

\begin{flalign}\label{eq43}
 W(\Phi_{i})\ =\ \mathcal{W}(\phi_{i}+\sqrt{2}\theta\xi_{i}+\theta\theta F_i)=\nonumber \\
 \displaystyle W(\phi_{i})+\frac{\partial \mathcal{W}}{\partial \phi_{i}}\sqrt{2}\theta\xi_{i}+\theta\theta(\frac{\partial \mathcal{W}}{\partial \phi_{i}}F_{i}-\frac{1}{2}\frac{\partial^{2}\mathcal{W}}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}), \nonumber \\
 \mathcal{L}_F|_{\theta\theta} = F_{i}^{\star}F_{i} + (\frac{\partial W}{\partial \phi_{i}}F_{i}+\frac{\partial W}{\partial \phi_{i}^{\star}}F_{i}^{\star}) -\frac{1}{2}\frac{\partial^{2}\mathcal{W}}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}, \nonumber \\
 \frac{\partial \mathcal{L}_F|_{\theta\theta}}{\partial F_i} = 0\Rightarrow F_i^{\star} = -\frac{\partial \mathcal{W}}{\partial \phi_{i}}, \nonumber \\
 \frac{\partial \mathcal{L}_F|_{\theta\theta}}{\partial F_i} = 0\Rightarrow F_i = -\frac{\partial \mathcal{W}}{\partial \phi_{i}^{\star}}, \nonumber \\
  F_i = -m\phi^{\star}_i - g\phi_i^{\star}\phi_i^{\star},\,\, F_i^{\star} = -m\phi_i - g\phi_i\phi_i.
\end{flalign}

\DUrole{raw-tex}{\noindent}
Making the substitution of \DUrole{raw-tex}{$F_i$}  derived  in  \DUrole{raw-tex}{$(\ref{eq43})$}, someone can infer that
SUSY Lagrangian \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$} describes   a Dirac massive fermion
field \DUrole{raw-tex}{$\xi(x)$}, a complex scalar field \DUrole{raw-tex}{$\phi(x)=1/\sqrt{2}(h(x)+iH(x))$} with scalar and pseudoscalar \DUrole{raw-tex}{$h,H$} as
mass eigenbasis of the Higgs sector determined by \DUrole{raw-tex}{$\phi(x)$}. The full SUSY Lagragngian of the model
\DUrole{raw-tex}{\cite{{Wess:1974jb},\cite{Wess:1974t}}  is  given by \DUrole{raw-tex}{$(\ref{eq44})$} in terms of the physical fields.
The Higgs sector is defined by the scalar potential \DUrole{raw-tex}{$V(h,H)$}
\DUrole{raw-tex}{$(\ref{eq45})$} having the mass  and self-interaction terms of the \DUrole{raw-tex}{$h$} and \DUrole{raw-tex}{$H$} fields.
The Yukawa interaction between  \DUrole{raw-tex}{$\xi$} fermion  and  \DUrole{raw-tex}{$\phi$} scalar fields
is given by  Lagrangian \DUrole{raw-tex}{$\mathcal{L}_I$} in  \DUrole{raw-tex}{$(\ref{eq44})$}. The other parts of SUSY Lagrangian \DUrole{raw-tex}{$\mathcal{L}_K$} and
\DUrole{raw-tex}{$\mathcal{L}_M$} are the kinetic and mass terms of the Lagrangian.

\begin{flalign}\label{eq44}
\mathcal{L}_K=\frac{1}{2} \partial _{\mu }[h]{}{}^2+\frac{1}{2} \partial _{\mu }[H]{}{}^2- \frac{1}{2} h \partial _{\mu }\left[\partial _{\mu }[h]\right]-\frac{1}{2} H \partial _{\mu }\left[\partial _{\mu }[H]\right]\nonumber \\
-\frac{1}{2} i \partial _{\mu }\left[\overset{-}{\xi }_{\alpha }\right].\xi _{\beta} \gamma {}^{\mu }.P^-_{\alpha ,\beta}+\frac{1}{2} i \overset{-}{\xi }_{\alpha }.\partial _{\mu }\left[\xi _{\beta }\right] \gamma {}^{\mu }.P^-_{\alpha ,\beta }-\nonumber \\
 \frac{1}{2} i \partial _{\mu }\left[\overset{-}{\xi }_{\beta }\right].\xi _{\alpha } \gamma {}^{\mu }.P^+_{\beta ,\alpha }+\frac{1}{2} i \overset{-}{\xi }_{\beta }.\partial _{\mu }\left[\xi _{\alpha }\right] \gamma {}^{\mu }.P^+_{\beta ,\alpha },\nonumber  \\
 \mathcal{L}_M=-\frac{1}{2} h{}^2 m{}^2-\frac{H{}^2 m{}^2}{2}-\frac{1}{4} m \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }-\nonumber \\
 \frac{1}{4} m \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }-\frac{1}{4} m \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }-\frac{1}{4} m \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^++_{\beta ,\alpha },\nonumber \\
 \mathcal{L}_I=-\frac{1}{4} g{}^2 h{}^4-\frac{1}{2} g{}^2 h{}^2 H{}^2-\frac{g{}^2 H{}^4}{4}-\frac{g h{}^3 m}{\sqrt{2}}-\frac{g h H{}^2 m}{\sqrt{2}}-\frac{g h \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }}{2 \sqrt{2}}\nonumber \\
-\frac{i g H \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }}{2 \sqrt{2}}-
\frac{g h \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }}{2 \sqrt{2}}-\frac{i g H \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }}{2 \sqrt{2}}\nonumber \\
-\frac{g h \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }}{2 \sqrt{2}}+\frac{i g H \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }}{2 \sqrt{2}}-\frac{g h \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^+_{\beta ,\alpha }}{2 \sqrt{2}}\nonumber \\
+\frac{i g H \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^+_{\beta ,\alpha }}{2 \sqrt{2}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Here \DUrole{raw-tex}{$\xi^C$} is the charge-conjugated field, and \DUrole{raw-tex}{$P^{\pm}$} denotes the left- and right-handed projective operators
\DUrole{raw-tex}{$$P^{\pm}=\frac{1_{4x4}\pm \gamma^5}{2}$$}

\begin{flalign}\label{eq45}
V(h,H)= \frac{g^2 h^4}{4}+\frac{1}{2} g^2 h^2 H^2+\frac{g^2
H^4}{4}+\frac{g h^3 m}{\sqrt{2}}+\frac{g h H^2
m}{\sqrt{2}}+\frac{h^2 m^2}{2}+\frac{H^2 m^2}{2}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The vacum states  of the quantum scalar field  system  are ones which satisfies
the minimum  of the scalar potential \DUrole{raw-tex}{$V(h,H)$} . The  solutions for such vacum expectation values  (v.e.v.s) of \DUrole{raw-tex}{$h,H$} are
shown in the equation \DUrole{raw-tex}{$(\ref{eq46})$}  and Fig. \DUrole{raw-tex}{$\ref{fig6}$} shows \DUrole{raw-tex}{$V(h,H)$} as a function  of  the scalar boson amplitude
\DUrole{raw-tex}{$|\phi|$} \DUrole{raw-tex}{$(\ref{eq47})$} .

 \begin{flalign}\label{eq46}
 h\to 0,H\to 0, \nonumber \\
 h\to -\frac{\sqrt{2} m}{g},H\to 0, \nonumber \\
 h\to 0,H\to \frac{\sqrt{2} m (\pm i)}{g}, \nonumber \\
\end{flalign}

\begin{flalign}\label{eq47}
\phi(x)=|\phi(x)|e^{i\cdot\alpha}, \,\, h(x)=Re(\phi(x)),\,\, H(x)=i\cdot Im(\phi(x))
\end{flalign}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoScalarPotential.png}}
\caption{The scalar potential \DUrole{raw-tex}{$V(h,H)$} as a function of the amplitude \DUrole{raw-tex}{$|\phi|$} and the phase \DUrole{raw-tex}{$\alpha$} of the Higgs field \DUrole{raw-tex}{\label{fig6}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Such non zero v.e.v.s of \DUrole{raw-tex}{$h,H$} when \DUrole{raw-tex}{$V$} has the minimum  is non-SUSY ground state which is  appeared
after spontaneous SUSY breaking.
This non-SUSY ground state  finally leads to the gauge symmetry  breaking ( \DUrole{raw-tex}{$U(1)$} in the case of the model
\DUrole{raw-tex}{\cite{Wess:1974jb}} , \DUrole{raw-tex}{\cite{Wess:1974tw}} ) breaking via the Higgs mechanism. On the other hand the SUSY breaking is resulting  from  auxiliary fields  \DUrole{raw-tex}{$F_i$} or
\DUrole{raw-tex}{$D_i$}  which occur to have non-zero v.e.v.s. as solutions of  the  equation of motion. There is an example of such \DUrole{raw-tex}{$F-$}
mechanish \DUrole{raw-tex}{\cite{O'Raifeartaigh:1975pr}}  shown in \DUrole{raw-tex}{$(\ref{eq48})$} .
In this case, the superpotential of the several chiral fields \DUrole{raw-tex}{$(\ref{eq48})$} doesn't provide
the   solutions consitent  with \DUrole{raw-tex}{$<0|F_i|0>=0$} for  some chiral supermutiplets and therefore SUSY is spontaneously
broken.

\begin{flalign}\label{eq48}
W(\Phi) = \lambda \Phi_3 + m\Phi_1\Phi_2 + g\Phi3\Phi_1^2 \Rightarrow \nonumber \\
F_1^{\star} = m\phi_2 + 2g\phi_1\phi_3,\,\,\,  F_2^{\star} = m\phi_1,\,\,\, F_3^{\star} = \lambda + g\phi_1^2
\end{flalign}

\DUrole{raw-tex}{\noindent}


\subsubsection{2.3.1~~~Soft-SUSY breaking%
  \label{soft-susy-breaking}%
}

The  described SUSY  model \DUrole{raw-tex}{\cite{Wess:1974jb},\cite{Wess:1974tw}} doesn't contain quadratic divergences destabilizing the scalar (higgs) sector.
Such divergences appear in the tadpoles and the selfenergies of the Higgs fields as it is illustrated by Fig. \DUrole{raw-tex}{$\ref{fig7},\ref{fig8}$}

% figure:  /mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoTadpole.PNG
% :width: 80%
% :align: center
% 
% Tadpole (one-point) contributions to the Higgs boson   `$h(x)$` masses at the one-loop level `\label{fig7}`
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.500\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoSelfEnergy.PNG}}
\caption{Two-point contributions of the fermion \DUrole{raw-tex}{$\xi$} and the scalar \DUrole{raw-tex}{$H(x)$}  to the Higgs boson \DUrole{raw-tex}{$h(x)$} mass \DUrole{raw-tex}{\label{fig8}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The one-loop contributions can be estimated. A simple calculation gives  the equations \DUrole{raw-tex}{$(\ref{eq49})$} in the
dimensional regularization \DUrole{raw-tex}{\cite{'tHooft:1972fi},\cite{Bollini:1972ui}, \cite{Bollini:1972ui}}. The loop integrals introduced in
\DUrole{raw-tex}{$(\ref{eq49})$}  are reduced to the scalar  one-point \DUrole{raw-tex}{$A_0(m^2)$} and two-point \DUrole{raw-tex}{$B_{0(1)}(p^2,m_1^2,m_2^2)$} functions
using tensor decomposition technique explained in \DUrole{raw-tex}{\cite{Passarino:1978jh}}.
The functions are ultra-violet (UV) divergent and their definitions can be found  in the \DUrole{raw-tex}{\cite{'tHooft:1978xw}} work..

\begin{flalign}\label{eq49}
 Tadpole\sim <0|\mathcal{L}_I|h>=-\frac{9 g^2 \text{A}_0\left(M_h^2\right) m^2}{16 M_h^2 \pi ^2} -\frac{3 g^2\text{A}_0(M_H^2) m^2}{16 M_h^2 \pi^2} + \frac{3 g^2\text{A}_0(M_{\xi}^2) m M_{\xi}}{4 M_h^2 \pi^2},\nonumber \\
 SelfEnergy\sim <h|\mathcal{L}_I|h>= \frac{3 A_0(M_h^2) g^2}{16 \pi ^2}+\frac{A_0(M_H^2) g^2}{16\pi^2}-\frac{A_0(M_{\xi}^2) g^2}{4 \pi ^2}+ \nonumber \\
 \frac{3 m^2  B_0(M_h^2,M_h^2,M_h^2) g^2}{16 \pi ^2}+\frac{m^2 B_0(M_h^2,M_H^2,M_H^2) g^2}{16 \pi^2} - \nonumber \\
-\frac{g^2 M_{\xi}^2 B_0(M_h^2,M_{\xi}^2,M_{\xi}^2)}{4 \pi^2}  - p_{\mu}\frac{M_h^2 g^2 B_1(M_h^2,M_{\xi}^2,M_{\xi}^2)}{4 \pi^2}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can explicitly show that \DUrole{raw-tex}{$(\ref{eq49})$} contains only \DUrole{raw-tex}{$\log (\Lambda^2/M_{h(H,\xi)}^2)$} divergent terms
\DUrole{raw-tex}{$(\ref{eq50})$} , if  \texttt{cut-off} regularization \DUrole{raw-tex}{\cite{Kleinert:2001hn},\cite{Polchinski:1983gv}} is applied to \DUrole{raw-tex}{$(\ref{eq49})$} . .

\begin{flalign}\label{eq50}
 A_0(m^2) \rightarrow m^2(\Lambda^2/m^2 - \log (\Lambda^2/m^2)),\,\, B_0(p^2,m^2,m^2)\rightarrow \log (\Lambda^2/m^2),\nonumber \\
 Tadpole=\frac{3 g^2 m } { 16 M_h^2 \pi^2} (-4 m \Lambda^2 + 4 M_{\xi}\Lambda^2     ,\nonumber \\
 + 3 M_h^2 m \log (\Lambda^2/M_h^2)  + M_H^2 m \log (\Lambda^2/M_H^2) - 4 M_{\xi}^3 \log (\Lambda^2/M_{\xi}^2), \nonumber \\
 SelfEnergy= \frac{g^2}{16 \pi^2}[(3m^2 - 3 M_h^2)\log (\Lambda^2/M_h^2) + (m^2 - M_H^2) \log (\Lambda^2/M_H^2)]
\end{flalign}

\DUrole{raw-tex}{\noindent}
The quadratic divergency is canceled only if the fermion mass \DUrole{raw-tex}{$M_{\xi}$}  is equal to  the parameter \DUrole{raw-tex}{$m$}
which determines  the mass term of the superpotential \DUrole{raw-tex}{$W(\Phi_i)$}. The case,  when the masses
of the scalars and fermions  after SUSY and gauge  symmetry breaking  are the same, \DUrole{raw-tex}{$M_H=M_h=M_{\xi}=m$} ,
corresponds to the total cancelation of the divergency in the one-loop effects.
Also it turns out that it is  possible to  extend  the superpotential \DUrole{raw-tex}{$W(\Phi_i)$} by terms \DUrole{raw-tex}{$\sim \lambda\Phi_i + m^{\prime}\Phi^2_i+ g^{\prime}\Phi^3_i $}
in such way that SUSY symmetry  is broken but only logarithmic divergences \DUrole{raw-tex}{$(\ref{eq50})$} will be presented at the one-loop level.
This is  the \texttt{soft-SUSY} breaking. MSSM is the theory with  the \texttt{soft-SUSY} breaking mechanism.


\section{3~~~The Minimal Supersymmetric Model%
  \label{the-minimal-supersymmetric-model}%
}

The SUSY theories has the qual number of bosonic and fermionic degrees of freedom. Minimal version of SUSY generalization  of
SM (MSSM) doubles the number of particles, introducing a superpartner to each particle \DUrole{raw-tex}{\cite{MSSM1}}. Unfortunately, the  \DUrole{raw-tex}{$F(D)-$} mechanisms like
\DUrole{raw-tex}{\cite{O'Raifeartaigh:1975pr}}  of SUSY breaking  don't explicitely work in the MSSM because any non-zero v.e.v.s for \DUrole{raw-tex}{$F-$} and
\DUrole{raw-tex}{$D-$}  terms of the SM fields spoil \DUrole{raw-tex}{$U(1)$} or \DUrole{raw-tex}{$SU(3)$} gauge invariance. Thus there is a hidden sector, the weakest part
of the MSSM, which is spontaneously breaking via \DUrole{raw-tex}{$F-$} mechanism. The special fields \texttt{messengers} like gravitino, gaugino or
gauge bosons can mediate SUSY breaking from hidden to the visible sector.

There are several requirements imposed on the Higgs sector that should be taken into account for the SUSY generalization of SM:
%
\begin{itemize}

\item Higgs fields have non-zero v.e.v.s. Therefore they cannot be superpartners of quarks  and leptons since
this would induce the spontaneous violation of baryon and lepton number through Yukawa interaction.

\item One needs at least two complex chiral multiplets to give masses to Up and Down quarks.

\end{itemize}

The latter is due to the form of the superpotential and chirality of the matter superfields. The Yukawa interaction in
Standard Model should be invariant under \DUrole{raw-tex}{$U(1)$} gauge subgroup. This imposes the fixed form of the interaction terms
\DUrole{raw-tex}{$$\mathcal{L}_Y = (y_e)_{\alpha\beta}\bar{L}_e^{\alpha} H E_e^{\beta} + (y_d)_{\alpha\beta}\bar{Q}^{\alpha} H D^{\beta} $$}
with zero hypercharge \DUrole{raw-tex}{$Y=-Y_L+Y_{H}+Y_E=-(-1)+1-2=0$}. The up quarks generates masses via interaction with charge-conjugated
Higgs doublet \DUrole{raw-tex}{$H^C=i\sigma^2H$} with \DUrole{raw-tex}{$Y_{H^C}=-1$}
\DUrole{raw-tex}{$$  \mathcal{L}_Y^u= (y_u)_{\alpha\beta}\bar{Q}^{\alpha}H^C U^{\beta},\,\,\, Y=-1/3-1+4/3=0.$$}
However, in SUSY, \DUrole{raw-tex}{$H$} is a left chiral superfield. Hence the superpotential, which is constructed out of the  left chiral
superfields, can contain only \DUrole{raw-tex}{$H$}  but not \DUrole{raw-tex}{$H^C$} which is the right chiral multiplet.

Another reason for introducing the  second left-handed Higgs doublet \DUrole{raw-tex}{$H_2$}  is the cancelation of chiral anomalies. The
chiral anomalies  \DUrole{raw-tex}{$Tr(Y^3)\neq 0$} spoils gauge invariance   and, hence,  the renormalizability of the theory.
SM is free of such chiral anomalies \DUrole{raw-tex}{$$ Tr(\sum_{i=e_L,e_R,\nu_L,u_L,etc} Y_i^3)=0 $$} but if a chiral Higgs doublet  is introduced,
it contains \texttt{higgsinos}, which are  fermions which hypercharges are non zero: \DUrole{raw-tex}{$Tr(Y^3) \neq 0$} . To compensate them,
one has to add the second Higgs doublet with opposite hypercharge.


\subsection{3.1~~~Field Content%
  \label{field-content}%
}

Minimal Supersymmetric Model associates Standard Model bosons with new fermions and SM fermions with new bosons.
The another Higgs  doublet is added to cancel the chiral anomalies and to make massive  up quarks. The particle
content of the MSSM  appears as \DUrole{raw-tex}{\cite{MSSM2}} illustrated in Table \DUrole{raw-tex}{$\ref{tbl2}$} .

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.114\DUtablewidth}|p{0.339\DUtablewidth}|p{0.197\DUtablewidth}|p{0.300\DUtablewidth}|}
\caption{Particle content of the MSSM \DUrole{raw-tex}{\label{tbl2}}}\\
\hline
\textbf{%
Superfield
} & \textbf{%
Bosons (spin 0,1)
} & \textbf{%
Fermions (spin 1/2)
} & \textbf{%
\DUrole{raw-tex}{$SU(3)_C$ \hfill $SU(2)_L$\hfill $U(1)_Y$}
} \\
\hline
\endfirsthead
\caption[]{Particle content of the MSSM \DUrole{raw-tex}{\label{tbl2}} (... continued)}\\
\hline
\textbf{%
Superfield
} & \textbf{%
Bosons (spin 0,1)
} & \textbf{%
Fermions (spin 1/2)
} & \textbf{%
\DUrole{raw-tex}{$SU(3)_C$ \hfill $SU(2)_L$\hfill $U(1)_Y$}
} \\
\hline
\endhead
\multicolumn{4}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

\textbf{Gauge}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$G_a$}
 & 
gluon, \DUrole{raw-tex}{$g_a$}
 & 
gluino, \DUrole{raw-tex}{$\tilde{g}_a$}
 & 
\DUrole{raw-tex}{8\hfill      0 \hfill       0}
 \\
\hline

\DUrole{raw-tex}{$V^k$}
 & 
weak,  \DUrole{raw-tex}{$W^k$}
 & 
wino, \DUrole{raw-tex}{$\tilde{W}^k$}
 & 
\DUrole{raw-tex}{1\hfill     3\hfill        0}
 \\
\hline

\DUrole{raw-tex}{$B$}
 & 
hypercharge, \DUrole{raw-tex}{$B$}
 & 
bino, \DUrole{raw-tex}{$\tilde{B}$}
 & 
\DUrole{raw-tex}{1\hfill     1\hfill        0}
 \\
\hline

\textbf{Matter}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$L_i$}
 & 
sleptons, \DUrole{raw-tex}{$(\tilde{\nu},\tilde{l}_L)$}
 & 
leptons, \DUrole{raw-tex}{$(\nu,l_L)$}
 & 
\DUrole{raw-tex}{1\hfill     2\hfill        -1}
 \\
\hline

\DUrole{raw-tex}{$E_i^C$}
 & 
\DUrole{raw-tex}{\hfill $\tilde{l}_R^C$ \hfill}
 & 
\DUrole{raw-tex}{\hfill $l_R^C$ \hfill}
 & 
\DUrole{raw-tex}{1\hfill     1\hfill        2}
 \\
\hline

\DUrole{raw-tex}{$Q_i$}
 & 
squarks, \DUrole{raw-tex}{$(\tilde{u},\tilde{d})_L$}
 & 
quarks, \DUrole{raw-tex}{$(u,d)_L$}
 & 
\DUrole{raw-tex}{3\hfill     2\hfill        1/3}
 \\
\hline

\DUrole{raw-tex}{$U_i^C$}
 & 
\DUrole{raw-tex}{$\tilde{u}_R^C$}
 & 
\DUrole{raw-tex}{$u_R^C$}
 & 
\DUrole{raw-tex}{$\bar{3}$\hfill     1\hfill       -4/3}
 \\
\hline

\DUrole{raw-tex}{$D_i^C$}
 & 
\DUrole{raw-tex}{$\tilde{d}_R^C$}
 & 
\DUrole{raw-tex}{$d_R^C$}
 & 
\DUrole{raw-tex}{$\bar{3}$\hfill     1\hfill       2/3}
 \\
\hline

\textbf{Higgs}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$H_1$}
 & 
Higgses, \DUrole{raw-tex}{$H_1$}
 & 
higgsinos, \DUrole{raw-tex}{$\tilde{H}_1$}
 & 
\DUrole{raw-tex}{1\hfill     $\bar{2}$\hfill        -1}
 \\
\hline

\DUrole{raw-tex}{$H_2$}
 & 
\DUrole{raw-tex}{$H_2$ \hfill {}}
 & 
\DUrole{raw-tex}{$\tilde{H}_2$\hfill {}}
 & 
\DUrole{raw-tex}{1\hfill     2\hfill        1}
 \\
\hline
\end{longtable}

The gauge (vector) superfields consist of the SM gauge bosons \DUrole{raw-tex}{$W^k,\,\, k=1,2,3$} and the gluon field \DUrole{raw-tex}{$g_a, \,\, a=1..8$}
accompanied  by their superpartners, spin 1/2 Majorana particles called \textbf{gauginos}  and gluino correspondingly.
Like the guage fields. these fermion fields transform as the adjoint representation of the appropriate group factor:
\DUrole{raw-tex}{$SU(3)_C,\,\, SU(2)_L\,\,\, U(1)_Y$} . Hereafter, tilde denotes a superpartner \DUrole{raw-tex}{$\tilde{l}$} of an ordinary particle, and
\DUrole{raw-tex}{$^C$} indicates on the charge-conjugation \DUrole{raw-tex}{$\psi_L^C = i\sigma_2\psi_R^{\star},\,\,\, \psi_R^C=-i\sigma_2\psi_L^{\star}$}

The chiral (matter) superfields content of the SUSY is exactly the same as in SM: three families of the chriral quarks and leptons.
Each family has five different gauge representations  of SM Weyl fermions \DUrole{raw-tex}{$\psi_L=(1/2,0),\,\, \psi_R=(0,1/2)$} shown as the
third column of the Table \DUrole{raw-tex}{\ref{tbl2}} :
\DUrole{raw-tex}{$$ Q_i =q_L(3,2)_{+1/3},\,\,U_i^C=u_R^C(\bar{3},1)_{-4/3},\,\, D_i^C=d_R^C(\bar{3},1)_{+2/3}, $$
$$L_i=l_L(1,2)_{-1},\,\, E_i^C=l_R^C(1,1)_{+2} $$}
In addition, the matter families in SUSY are populated  by five gauge representation of spin-0 particles,
scalars and pseudoscalars, with
the same quantum numbers of the electric charge \DUrole{raw-tex}{$Q$} and third projection of the weak isospin \DUrole{raw-tex}{$T_3$} as their SM partners have
\DUrole{raw-tex}{$$scalar(pseudo)\sim(0,1/2)\times (1/2,0)\sim \overline{\psi}\psi(\overline{\psi}\gamma^5\psi),$$} where
\DUrole{raw-tex}{$$ scalar=\psi_R^{\dag}\psi_L + \psi_L^{\dag}\psi_R,\,\,\,\,\, pseudo=\psi_R^{\dag}\psi_L - \psi_L^{\dag}\psi_R $$}
The scalar partners \DUrole{raw-tex}{$\tilde{l}_L$} and  \DUrole{raw-tex}{$\tilde{l}_R$} couples only to \DUrole{raw-tex}{$\tilde{l}_L\,\,(l_L)$} and
\DUrole{raw-tex}{$\tilde{l}_R\,\,(l_R)$} via gauge boson (gauginos) because the gauge interaction  conserves the chiral flavor.
However, there can be \DUrole{raw-tex}{$\tilde{l}_L-\tilde{l}_{R}$} mixing in MSSM, if one considers the  Yukawa interaction
in the superpotential which violates \DUrole{raw-tex}{$R-$} parity. Such terms in SM would violate fermion and
barion numbers. This mixing is generally negligible except for third generation of sfermions.

The presence of an extra Higgs doublet in SUSY model is a novel feature of the SM-like theory. In the MSSM, one has two doublets
\DUrole{raw-tex}{$H_1$} and \DUrole{raw-tex}{$H_2$} of the representions \DUrole{raw-tex}{$(1,2,-1)$} and \DUrole{raw-tex}{$(1,\bar{2},+1)$} accordingly
\DUrole{raw-tex}{$$ H_1=\left(\begin{array}{l}  H_1^0 & \\
H_1^- \end{array} \right)=\left( \begin{array}{l} v_1 + \frac{S_1^0(x) + iP_1^0(x)}{\sqrt{2}}  & \\ \frac{S_1^-(x) + iP_1^-(x)}{\sqrt{2}} & \end{array} \right),\,\,
H_2=\left(\begin{array}{l}
H_2^+ & \\
H_2^0 & \end{array} \right)=\left( \begin{array}{l}  \frac{S_2^+(x) + iP_2^+(x)}{\sqrt{2}} & \\ v_2 + \frac{S_2^0(x) + iP_2^0(x)}{\sqrt{2}}  &  \end{array} \right),\,\,
$$}
If the vacum state is not  \DUrole{raw-tex}{$SU(2)_L$} invariant any more, the three degrees
of freedom \DUrole{raw-tex}{$P_1^0,P_1^-,P_2^+$}
are the Goldston modes \DUrole{raw-tex}{\cite{Goldstone:1961eq}, \cite{Goldstone:1962es}} which can
be gauged away \DUrole{raw-tex}{\cite{LopezOsorio:2004pu}} using unitary gauge  \DUrole{raw-tex}{\cite{Weinberg:1973ew}} .
Hence there are only \DUrole{raw-tex}{$5=8-3$}
dynamical degrees of  freedom resulting into the five  massive physical states
\DUrole{raw-tex}{$$ h(x)\sim S_1^0, \,\,\, H(x)\sim S_2^0,\,\,\, H^-(x)\sim S_1^-,\,\,\, H^+(x)\sim S_2^+,\,\,\, A(x)\sim P_2^0(x),  $$}
where \DUrole{raw-tex}{$h(x)$} and \DUrole{raw-tex}{$H(x)$}  are CP even neuthral Higgs bosons, \DUrole{raw-tex}{$A(x)$} is a CP odd neutral Higgs boson and \DUrole{raw-tex}{$H^{\pm}(x)$} are
two charged Higgs fields. I will consider the mass eigenstates of the Higgs sector below.


\subsection{3.2~~~Lagrangian of the MSSM%
  \label{lagrangian-of-the-mssm}%
}

If supersymmetry is exact, superpartners of ordinary particles have the same masses and have to be observed. The absence of
them at modern energies is believed to be explained by the fact their masses are very heavy, that means that the supersymmetry
should be broken. Hence, if the energy of accelerators would be  high enough, the superpartners could be created during
particle collisions.

The Lagrangian of the MSSM consists of two parts. The first part is SUSY generalization of the Standard Model, while
the second one represents the SUSY breaking.  Recalling \DUrole{raw-tex}{$(\ref{eq37})$} and extending the gauge group to
\DUrole{raw-tex}{$SU(3)_C\times SU(2)_L\times U(1)_Y$} \DUrole{raw-tex}{$$ \Phi^{\dag}e^{gV}\Phi \Rightarrow \Phi^{\dag}e^{g_3V_3+g_2V_2+g_1V_1}\Phi $$}
someone can  write

\begin{flalign}\label{eq51}
\mathcal{L} = \mathcal{L}_{SUSY}  + \mathcal{L}_{breaking},\nonumber \\
\mathcal{L}_{SUSY} = \mathcal{L}_{gauge} + \mathcal{L}_{Yukawa},\nonumber \\
\mathcal{L}_{gauge} = \sum_{SU(3),SU(2),U(1)}\frac{1}{4}(\int d^{2}\theta TrW^{\alpha}W_{\alpha}+\int d^{2}\overline{\theta}Tr\overline{W}^{\alpha}\overline W_{\alpha}) + \nonumber \\
\sum_{matter}\int d^{2}\theta d^{2}\overline{\theta}\Phi_{i}^{\dagger}e^{g_{3}\hat{V}_{3}+g_{2}\hat{V}_{2}+g_{1}\hat{V}_{1}}\Phi_{i},\nonumber \\
\mathcal{L}_{Yukawa} = \int d^{2}\theta \mathcal{W}_{R}+h.c
\end{flalign}

\DUrole{raw-tex}{\noindent}
In terms of the component fields, the Lagrangians \DUrole{raw-tex}{$\mathcal{L}_{gauge}$}  and \DUrole{raw-tex}{$\mathcal{L}_{Yukawa}$} introduced  above  \DUrole{raw-tex}{$(\ref{eq51})$} take a familiar form

\begin{flalign}\label{eq52}
\mathcal{L}_{gauge} = \sum_{a=SU(3),SU(2),U(1)}(-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}-i\lambda^{a}\sigma^{\mu}D_{\mu}\overline{\lambda}^{a}-\frac{1}{2}D^{a}D^{a} +   \nonumber \\
 D_{\mu}\phi_i^{\star}D^{\mu}\phi_i + i \xi^{\dag}\sigma^{\mu}D_{\mu} \xi - i\sqrt{2}g^a ( \phi_i^{\star}  T^a\phi_i \lambda_a^T\xi_i - \xi^{\dagger}_iT^a\lambda_a^{\star}\phi_i)   \nonumber \\
 \mathcal{L}_{Yukawa}=-F_i^{\dagger}F_i  - (\frac{1}{2} \xi_{i}^T \frac{ \partial^2  \mathcal{W}_R } {\partial \phi_i\partial \phi_j}\xi_j + h.c.)
\end{flalign}

\DUrole{raw-tex}{\noindent}
The equations of motions for  the auxilary fields \DUrole{raw-tex}{$F_i$} and \DUrole{raw-tex}{$D_a$}
\DUrole{raw-tex}{$$ F_i^{\star}=-\frac{\partial \mathcal{W}_R} {\partial \phi_i},\,\,\, D_a=-g^a\phi_i(T^a)_{ij}\phi_j $$}
were used to derive \DUrole{raw-tex}{$(\ref{eq52})$} .
I will focus on the SUSY generalization of Yukawa interaction \DUrole{raw-tex}{$\mathcal{W}_R$} which preserves the \DUrole{raw-tex}{$R-$} parity:

\begin{flalign}\label{eq53}
\mathcal{W}_{R}=\epsilon_{ij}(y_{ab}^{U}Q_{a}^{j}U_{b}^{C}H_{2}^{i}+y_{ab}^{D}Q_{a}^{j}D_{b}^{C}H_{1}^{i}+y_{ab}^{L}L_{a}^{j}E_{b}^{C}H_{1}^{i}+\mu H_{1}^{i}H_{2}^{j}),
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$y^{U,D,L}_{ab}$} are the \DUrole{raw-tex}{$3\times 3$} matrices  and \DUrole{raw-tex}{$Q_i$, $L_i$, $U^C_i$, $D_i^C$} are the supermultiplets given by the Table \DUrole{raw-tex}{${\ref{tbl2}}$} .
It is easy to show using \DUrole{raw-tex}{$(\ref{eq33})$, $(\ref{eq34})$ , $(\ref{eq366})$} that  \DUrole{raw-tex}{$R-$} parity invariance of \DUrole{raw-tex}{$\mathcal{W}_R$}
requires to have for any terms in the superpotential  \DUrole{raw-tex}{$$ R_p(W_R)=(-1)^{2k}, $$}  where \DUrole{raw-tex}{$k$} is some  integer
number. This leads to the following phenomenological predictions:
%
\begin{itemize}

\item Sparticles  can only be paired produced

\item Sparticles decay to SM particles and an odd number of sparticles

\item A chain of sparticles decays must be finalized by the production of the  stable Lightest Supersymmetric Particle (LSP)

\end{itemize}


\subsection{3.3~~~Breaking SUSY in the MSSM%
  \label{breaking-susy-in-the-mssm}%
}

As I mentioned previously, SUSY of MSSM can't be broken by developing non-zero v.e.v.s of some fields, i.e. the auxilary \DUrole{raw-tex}{$F-$}  fields,
without spoiling the gauge invariance. The most common scenario for producing low-energy supersymmetry breaking is   the
mediation of the  breaking from the hidden sector by means of the massive messengers \DUrole{raw-tex}{\cite{SUSYbreaking}} \DUrole{raw-tex}{$X$} with \DUrole{raw-tex}{$$M_X\sim <0|F_X|0> \sim M_{SUSY}\times M_{Pl}\sim 10^{22} GeV$$}
The effective theory of SUSY with the hidden sector contains higher dimension operators \DUrole{raw-tex}{$W \sim  X^{\dag}X\Phi_i^{\dag}\Phi_i$}
\DUrole{raw-tex}{$$ \Phi_i^{\dag}\Phi_i + X^{\dag}X+\frac{c_i}{M_{Pl}}X^{\dag}X\Phi_i^{\dag}\Phi_i,$$}
which give mass terms of the chiral superfield \DUrole{raw-tex}{$\Phi_i$} when all heavy states \DUrole{raw-tex}{$X$} are integrated out:

\begin{flalign}\label{eq54}
M_X^2\sim <0|F_X|0>,\,\,\,\frac{c_i}{M_{Pl}}X^{\dag}X\Phi_i^{\dag}\Phi_i  \Rightarrow M_{\Phi_i}^2 \Phi_i^{\dag}\Phi_i,\,\,\, M_{\Phi_i}^2 = c_i \frac{M_X}{M_{Pl}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The SUSY breaking \DUrole{raw-tex}{$$ <0|F_X|0>\neq 0 $$} in the hidden sector is propogated  to the visible sector, MSSM, and breaks it.
The   messenger  \DUrole{raw-tex}{$X$} may be a spin 3/2 particle, \emph{gravitino}  in the gravity mediation (\textbf{SUGRA})
breaking scenario \DUrole{raw-tex}{\cite{SUGRA}}. Or \DUrole{raw-tex}{$X$} might be a SM-singlet with non-zero v.e.v.s which couples to the chiral superfields \DUrole{raw-tex}{$Q_3$} and  \DUrole{raw-tex}{$\bar{Q}_3$} to be
\DUrole{raw-tex}{3 and $\bar{3}$} representations of  SU(3) \DUrole{raw-tex}{$$ W\sim X\bar{Q}_3Q_3.$$}    \DUrole{raw-tex}{$Q_3 (\bar{Q}_3)$} chiral multiplets become
the messengers dynamically breaking the supersymmetry at the low energy scales \DUrole{raw-tex}{$<F_X> << M_{Pl}$}. Masses of the gauginos  and
sfermions in the scenario of gauge mediated SUSY breaking (\textbf{GMSB} ) are the one-loop and two-loop effects initiated by
\DUrole{raw-tex}{$Q_3(\bar{Q}_3)$} messengers.

These mechanisms  can be  phenomenologicaly incorporated  into the MSSM  as some extensions of the Lagrangian
via the effective high dimension operator \DUrole{raw-tex}{$\mathcal{L}_{soft}$} after integration of  all messengers \DUrole{raw-tex}{$X$} out.

.

\begin{flalign}\label{eq55}
\mathcal{L}_{soft}= -\sum_{i}M_{i}^{2}|\phi_{i}|^{2}-(\sum_{\alpha}M_{\alpha}\tilde{\lambda}^C_{\alpha}\tilde{\lambda}_{\alpha}-B\mu\epsilon_{ab}H^a_{1}H^b_{2} \nonumber \\
 A_{ab}^{U}\tilde{Q}_{a}\tilde{U}_{b}^{c}H_{2}+A_{ab}^{D}\tilde{Q}_{a}\tilde{D}_{b}^{c}H_{1}+A_{ab}^{L}\tilde{L}_{a}\tilde{E}_{b}^{c}H_{1}+h.c.)
\end{flalign}

\DUrole{raw-tex}{\noindent}
The bilinear and trilinear couplings \DUrole{raw-tex}{$B$ and $A_{ab}$} are such that gauge invariance is not broken. The Lagrangian \DUrole{raw-tex}{$(\ref{eq54})$}
is only possible choice that does not destroy the renormalizability of the theory \DUrole{raw-tex}{\cite{renorm}}.
The terms \DUrole{raw-tex}{$(\ref{eq54})$} provide the mass splitting between particles living in
the same supermultiplet and ensure the high  masses of
sparticles. \DUrole{raw-tex}{$\mathcal{L}_{soft}$}  introduces 104 additional parameters which are purely phenomenological.
I'll focus on the Higgs
sector of MSSM and show how  Electroweak Symmetry breaking (\textbf{EW})   is dynamically achieved
with the help of Renormalization Group
(\textbf{RG}) equations of the Higgs sector.


\subsection{3.4~~~The Higgs sector and Electroweak Symmetry Breaking in the MSSM%
  \label{the-higgs-sector-and-electroweak-symmetry-breaking-in-the-mssm}%
}

The Higgs potential in the MSSM is fully defined by the superpotential \DUrole{raw-tex}{$(\ref{eq53})$} and   the soft-symmetry breaking
terms \DUrole{raw-tex}{$(\ref{eq55})$}. The Higgs potential consists of the  F-terms and D-terms of the SUSY Lagrangian \DUrole{raw-tex}{$(\ref{eq51})$}
The F-terms  come from  the soft-breaking terms \DUrole{raw-tex}{$(\ref{eq55})$}  and
contribute only to diagonal elements of the Higgs mass matrix in the gauge basis, while
D-terms are derived from Yukawa superpotential`\$(ref\{53\})\$` and rise the self-interaction in the Higgs sector.
The Higgs potential at tree level is

\begin{flalign}\label{eq56}
 V(H_{1},\ H_{2})\ =\ m_{1}^{2}|H_{1}|^{2}+m_{2}^{2}|H_{2}|^{2}-m_{3}^{2}(H_{1}H_{2}+h.c.) +  \nonumber \\
 \frac{g_2^{2}+g_1^{2}}{8}(|H_{1}|^{2}-|H_{2}|^{2})^{2}+\frac{g_2^{2}}{2}|H_{1}^{+}H_{2}|^{2},
 \end{flalign}

\DUrole{raw-tex}{\noindent}
where I've introduced the notation

\begin{flalign}\label{eq57}
m_{1}^2 = M_{H_1}^2 + \mu^2,\nonumber \\
m_{2}^2 = M_{H_2}^2 + \mu^2,\nonumber \\
m_{3}^2 = -B\mu
\end{flalign}

\DUrole{raw-tex}{\noindent}
There are the gauge couplings \DUrole{raw-tex}{$g_2$} and \DUrole{raw-tex}{$g_1$} of \DUrole{raw-tex}{$SU(2)$} and \DUrole{raw-tex}{$U(1)$} symmetry groups accordingly.
The parameters \DUrole{raw-tex}{$M_{H_i}$}, \DUrole{raw-tex}{$B$} in  \DUrole{raw-tex}{$(\ref{eq56})$}, \DUrole{raw-tex}{$(\ref{eq57})$}    are described by the  soft-symmetry breaking
Lagrangian \DUrole{raw-tex}{$\matcal{L}_{soft}$} \DUrole{raw-tex}{$(\ref{eq55})$}.

The potential \DUrole{raw-tex}{$(\ref{eq56})$} of the Higgs sector  is positive defined  with minima defined by the tadpole equations
\DUrole{raw-tex}{$(\ref{eq58})$}

\begin{flalign}\label{eq58}
 \frac{1}{2}\frac{\partial V}{\partial \S_{2}} = \frac{1}{8} \Big(-4 v_1 \Big(B + B^*\Big) + \Big(g_{1}^{2} + g_{2}^{2}\Big)v_{2}^{3}  + v_2 \Big(8 m_{H_2}^2  + 8 |\mu|^2  - \Big(g_{1}^{2} + g_{2}^{2}\Big)v_{1}^{2} \Big)\Big)=0,\nonumber\\
\frac{1}{2}\frac{\partial V}{\partial \S_{1}} = \frac{1}{8} \Big(-4 v_2 \Big(B+B^{*}\Big)  + v_1 \Big(8 m_{H_1}^2  + 8 |\mu|^2  - \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(- v_{1}^{2}  + v_{2}^{2}\Big)\Big)\Big)=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
We can parametrize  v.e.v.s of Higgs doublets \DUrole{raw-tex}{$v_1$}  and \DUrole{raw-tex}{$v_2$}  as \DUrole{raw-tex}{$$ v_1=v\times \cos\beta,\,\, v_2=v\times \sin\beta, $$}
introducing  a real positive parameter \DUrole{raw-tex}{$v$} which determines the mass \DUrole{raw-tex}{$m_A$}  of the CP-odd  \DUrole{raw-tex}{$A(x)$} neutral Higgs boson.
The parameter \DUrole{raw-tex}{$\beta$} or \DUrole{raw-tex}{$tan \beta$} determines Yukawa, gauge and  self interactions, i.e.
the couplings of the Higgs bosons to the fermions, vector bosons and scalars.

The solution of the equations \DUrole{raw-tex}{$(\ref{eq58})$} can be expressed in terms of \DUrole{raw-tex}{$v^2$} and \DUrole{raw-tex}{$\sin 2\beta$}

\begin{flalign}\label{eq59}
 v^{2}=\displaystyle \frac{4(m_{1}^{2}-m_{2}^{2}\tan^{2}\beta)}{(g^{2}+g^{\prime 2})(\tan^{2}\beta-1)}, \nonumber \\
\sin 2\beta=\frac{2m_{3}^{2}}{m_{1}^{2}+m_{2}^{2}}.
\end{flalign}

\DUrole{raw-tex}{\noindent}
The minimum of the Higgs potentional exists if  the two criteria are fulfilled   \DUrole{raw-tex}{$$v^2 > 0,\,\,\, |\sin 2\beta|\leq 1$$}
what translates to the following inequations

\begin{flalign}\label{eq60}
m_{1}^{2}+m_{2}^{2}>2m_{3}^{2},\nonumber \\
m_{1}^{2}m_{2}^{2}<m_{3}^{4}.
\end{flalign}

\DUrole{raw-tex}{\noindent}
It is clear that the relations \DUrole{raw-tex}{$(\ref{eq60})$} are not satisfied at the GUT scale when \DUrole{raw-tex}{$v^2$} happens to be negative
\DUrole{raw-tex}{$$ m_1^2=m_2^2=m_0^2+\mu_0^2,\,\, v_0^2<0$$}.
However the spontaneous breaking \DUrole{raw-tex}{$SU(2)$} is possible, if  the energy scale \DUrole{raw-tex}{$Q$} is running from GUT
\DUrole{raw-tex}{$Q\sim 10^{16}\,\, GeV$} level to some EWSB \DUrole{raw-tex}{$Q\sim 10^2\,\, GeV$} level. The parameters of the superpotentional
become \texttt{running} parameters with energy dependencies given by Renormalization Group (\textbf{RG}). This running Higgs masses
can change their positive signs when running down to \DUrole{raw-tex}{$Q_{EWSB}$} scale due to the radiative corrections in such way
that the scalar potential will have a non-trivial solution. Indeed, the Figure \DUrole{raw-tex}{$\ref{fig9}$} shows the behavior of
\DUrole{raw-tex}{$m_1^2$} and  \DUrole{raw-tex}{$m_2^2$} predicted by the renormalization group evolution   with 2-loop MSSM  \DUrole{raw-tex}{$\beta$} functions  in  the \DUrole{raw-tex}{$3^{rd}$}
generation approximation for SUSY parameters \DUrole{raw-tex}{\cite{ref1}} , \DUrole{raw-tex}{\cite{ref2}}. The Figure  \DUrole{raw-tex}{$\ref{fig10}$} presents
v.e.v.s RGE, where  the region of the positive \DUrole{raw-tex}{$v^2$} correspondes to the area filled with  gray color in Figure \DUrole{raw-tex}{$\ref{fig9}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/MH1_MH2_RGE_v2.png}}
\caption{The running soft SUSY breaking parameters \DUrole{raw-tex}{$m_1$} and \DUrole{raw-tex}{$m_2$} parameters in the  SPS1  scenario  \DUrole{raw-tex}{\label{fig9}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The \texttt{SPS1} \DUrole{raw-tex}{\cite{ref3}}
benchmark point  for \texttt{mSUGRA}  MSSM scenario was taken as the initialial condition for  the numerical solving
two-loop MSSM RGE.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/vevs_RGE_v2.png}}
\caption{The running \DUrole{raw-tex}{$v^2$},  Higgs v.e.v.s    parameter in the  SPS1 scenario  \DUrole{raw-tex}{\label{fig10}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The Figure \DUrole{raw-tex}{$\ref{fig9}$} explicitely shows the order of \DUrole{raw-tex}{$M_{SUSY}$}at GUT  \DUrole{raw-tex}{$Q_{GUT}$} scale
when \DUrole{raw-tex}{$m_i\sim M_{H_i}\sim M_{SUSY}$}
\DUrole{raw-tex}{$M_{SUSY}$} is  \DUrole{raw-tex}{$10^2\div 10^3$}. Due to the logarithmic running of \DUrole{raw-tex}{$m_1$}  and \DUrole{raw-tex}{$m_2$},
one needs a long 'running time'
to  get \DUrole{raw-tex}{$m_2^2$} to be negative for the spontaneous  Electroweak Symmetry breaking  (\textbf{EWSB}).


\subsubsection{3.4.1~~~The Higgs sector of  the MSSM after EWSB%
  \label{the-higgs-sector-of-the-mssm-after-ewsb}%
}

The running of parameters leads to radiative EWSB. Provided conditions \DUrole{raw-tex}{$(\ref{eq60})$}
of EWSB are satisfied, the physical spectrum of the Higgs sector in MSSM can be obtained
via finding the eigenstates of the mass matrices for Higgs fields.

The mass matrix of CP-odd components \DUrole{raw-tex}{$P_1$}  and \DUrole{raw-tex}{$P_2$} in the gauge basis at the tree level is

\begin{flalign}\label{eq61}
m^2_{CP-odd} = \frac{\partial^2 V}{\partial P_i \partial P_j} =  \left(
\begin{array}{cc}
\frac{1}{8} \Big(8 m_{H_1}^2  + 8 |\mu|^2  + \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(- v_{2}^{2}  + v_{1}^{2}\Big)\Big) &{\Re\Big(B\mu\Big)}\\
{\Re\Big(B\mu\Big)} &m_{22}\end{array}
\right),  \nonumber \\
m_{22} = \frac{1}{8} \Big(8 m_{H_2}^2  + 8 |\mu|^2  - \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(- v_{2}^{2}  + v_{1}^{2}\Big)\Big)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
This matrix is diagonalized by a rotation matrix \DUrole{raw-tex}{\(Z^A\)}

\begin{flalign}\label{eq62}
 Z^A m^2_{CPP-odd} Z^{A,\dagger} = m^{diag,2}_{A^0},\nonumber \\
 G^0 = \sum_{j}Z_{{j 1}}^{A}P_{{j}}\,, \hspace{1cm}
 A^0 = \sum_{j}Z_{{j 2}}^{A}P_{{j}}\,\nonumber \\
 Z^A =   \left(
 \begin{array}{cc}
 -cos\beta & sin\beta\\
 sin\beta & cos\beta\end{array}
 \right),
 \end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$G^0$} is a Goldstone boson gauged away providing the longitudinal polarization of the massive
\DUrole{raw-tex}{$Z-$}boson. The mass and rotation matrices of the charged components \DUrole{raw-tex}{$H^-$} and \DUrole{raw-tex}{$H^+$}   have
the form

\begin{flalign}\label{eq63}
 m^2_{charged} =\frac{\partial^2 V}{\partial H^{+}_i \partial H^{-}_j}  =  \left(
 \begin{array}{cc}
  m_{11} &\frac{1}{4} g_{2}^{2} v_1 v_2  + B\mu^*\\
  \frac{1}{4} g_{2}^{2} v_1 v_2  + B\mu &m_{22}\end{array}
   \right),\nonumber \\
  m_{11} = \frac{1}{8} \Big(8 m_{H_1}^2  + 8 |\mu|^2  + g_{1}^{2} \Big(- v_{2}^{2}  + v_{1}^{2}\Big) + g_{2}^{2} \Big(v_{1}^{2} + v_{2}^{2}\Big)\Big)\nonumber \\
  m_{22} = \frac{1}{8} \Big(8 m_{H_2}^2  + 8 |\mu|^2  + \Big(- g_{1}^{2}  + g_{2}^{2}\Big)v_{1}^{2}  + \Big(g_{1}^{2} + g_{2}^{2}\Big)v_{2}^{2} \Big) \nonumber \\
  Z^+ m^2_{H^-} Z^{+,\dagger} = m^{diag,2}_{H^-},\nonumber \\
          Z^+ =   \left(
 \begin{array}{cc}
 -cos\beta & sin\beta\\
 sin\beta & cos\beta\end{array}
 \right),
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The mass matrix of CP-even neutral components \DUrole{raw-tex}{$S_1$}  and \DUrole{raw-tex}{$S_2$} can be written as

\begin{flalign}\label{eq64}
m^2_{h} = \frac{\partial^2 V}{\partial S_i \partial S_j}  = \left(
\begin{array}{cc}
 m_{11} &-\frac{1}{4} \Big(g_{1}^{2} + g_{2}^{2}\Big)v_1 v_2  - {\Re\Big(B\mu\Big)} \\
 -\frac{1}{4} \Big(g_{1}^{2} + g_{2}^{2}\Big)v_1 v_2  - {\Re\Big(B\mu\Big)}  &m_{22}\end{array}
 \right),\nonumber \\
     m_{11} = \frac{1}{8} \Big(8 m_{H_1}^2  + 8 |\mu|^2  + \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(3 v_{1}^{2}  - v_{2}^{2} \Big)\Big),\nonumber \\
  m_{22} = \frac{1}{8} \Big(8 m_{H_2}^2  + 8 |\mu|^2  - \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(-3 v_{2}^{2}  + v_{1}^{2}\Big)\Big),\nonumber \\
  Z^{H} m^2_{h} Z^{H,\dagger} = m^{diag,2}_{h},\nonumber \\
  h = \sum_{j}Z_{{j 1}}^{H}S_{{j}}\,, \hspace{1cm}
  H = \sum_{j}Z_{{j 2}}^{H}S_{{j}},
 \end{flalign}

\DUrole{raw-tex}{\noindent}
where the rotation maxtrix \DUrole{raw-tex}{$Z^{H}$} is given by

\begin{flalign}\label{eq65}
   Z^H =   \left(
 \begin{array}{cc}
 -cos\alpha & sin\alpha\\
 sin\alpha & cos\alpha\end{array}
 \right),\nonumber \\
 tan 2\alpha = - tan 2\beta \left(\frac{m_A^2+M_Z^2}{m_A^2-M_Z^2} \right).
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The masses of CP-odd neutral Higgs boson \DUrole{raw-tex}{$m_A$}, CP-even neutral  Higgses \DUrole{raw-tex}{$m_H,m_h$} and the
charged Higgses \DUrole{raw-tex}{$m_{H^{\pm}}$}  acquire the following definitions at the tree level:

\begin{flalign}\label{eq66}
 m_A^2 = m_1^2 + m_2^2,\nonumber\\
 m_{H,h}^2 = \frac{1}{2}(m_A^2 + M_Z^2\pm \sqrt{(m_A^2+M_Z^2)^2-4 m_A^2 M_Z^2 cos^2 2\beta}),\nonumber \\
 m_{H^{\pm}}^2 = m_A + M_{W^{\pm}}^2
\end{flalign}

\DUrole{raw-tex}{\noindent}
The equations \DUrole{raw-tex}{$(\ref{eq66})$} lead to MSSM Higgs masses relations

\begin{flalign}\label{eq67}
 m_{H^{\pm}}\geq M_{W}, \nonumber \\
 m_{h}\leq m_{A}\leq M_{H}, \nonumber \\
 m_{h}\leq M_{Z}|\cos 2\beta|\leq M_{Z}, \nonumber \\
m_{h}^{2}+m_{H}^{2}=m_{A}^{2}+M_{Z}^{2}.
\end{flalign}

\DUrole{raw-tex}{\noindent}
From the equation \DUrole{raw-tex}{$(\ref{eq67})$}  it follows that at the tree level the mass of the lightest
neutral Higgs boson, \DUrole{raw-tex}{$h$} is smaller the mass of \DUrole{raw-tex}{$Z-$}boson which clearly distinguishes it from
SM Higgs boson. It should have been observed at LEP2 \DUrole{raw-tex}{\cite{42 hep-ph05031173}}.  It had seemes
that the failure of detecting  the Higgs boson at LEP would have ruled out the MSSM as a viable theory for
physics at the weak scale. However, it was realized, first in \DUrole{raw-tex}{\cite{13 hep-ph 04061166v3}}
radiative corrections, in particular from top-quark and scalar top-quark loops,
could considerably  increase the mass of the lightest Higgs bosons to  a value beyond the
reach of LEP2. The upprer bound on \DUrole{raw-tex}{$m_h$} can be pushed up to about 135 GeV \DUrole{raw-tex}{\cite{hep-ph05031173}},
\DUrole{raw-tex}{\cite{13 hep-ph1112.3028v3}}, if one includes these one-loop \DUrole{raw-tex}{$\mathcal{Q}(\alpha)$} corrections in
the Higgs sector of MSSM.


\subsection{3.5~~~Radiative corrections and the upper bound on \DUrole{raw-tex}{$ M_h$}%
  \label{radiative-corrections-and-the-upper-bound-on-m-h}%
}

The equation \DUrole{raw-tex}{$(\ref{eq67})$}   shows that the lightest  CP-even Higgs boson should have
a mass below \DUrole{raw-tex}{$M_Z$} at the tree level \DUrole{raw-tex}{\cite{12 hep-ph0406166v3}}.
The upper bound on \DUrole{raw-tex}{$M_h$} \DUrole{raw-tex}{$$ M_h \sim M_Z $$} is reached when the mass \DUrole{raw-tex}{$M_A$}
of the CP-odd \DUrole{raw-tex}{$A$}
neutral Higgs boson is much larger then  the  \DUrole{raw-tex}{$M_Z$} mass of Z-boson and \DUrole{raw-tex}{$\beta \rightarrow \pi/2$}.
The saturation of \DUrole{raw-tex}{$M_h$} is shown in the Figure \DUrole{raw-tex}{$\ref{fig11}$}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/mSUGRA_Mh_vs_MA.png}}
\caption{The mass of the lightest Higgs boson \DUrole{raw-tex}{$h$},  \DUrole{raw-tex}{$M_h$} as a function of \DUrole{raw-tex}{$M_A$}
and \DUrole{raw-tex}{$tan\beta$} in mSUGRA MSSM. \DUrole{raw-tex}{\label{fig11}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
In the limits of the large \DUrole{raw-tex}{$M_A$} and \DUrole{raw-tex}{$tan\beta$} that are considered for the upper bound on \DUrole{raw-tex}{$M_h$},
the radiatuve effects are mainly provided by one- or two-point  contributions of the top and stop loops
if one assumes that all squarks have the same mass \DUrole{raw-tex}{$$m_{\tilde{t}}=m_{\tilde{b}}=M_S$$},
there is no mixing  of left- and right-components of squarks in the mass matrix \DUrole{raw-tex}{$$X_t=A_t-\mu cot\beta\ll M_S$$}
and the mass of the \DUrole{raw-tex}{$h$} Higgs boson \DUrole{raw-tex}{$M_h$}  is negligible \DUrole{raw-tex}{$M_h\ll M_S$} to skip the contribution
\DUrole{raw-tex}{$ \sim p_\mu M_h^2B_1(M_h^2,m_{\tilde{t}}^2,m_{\tilde{t}}^2)$}  from the self-energy  \DUrole{raw-tex}{$(\ref{eq49})$}
then the one-loop radiative correction reads

\begin{flalign}\label{eq68}
 \Delta M_h^2 =  \frac{3 G_F}{\sqrt{2} \pi^2} m_t^4 \, {\rm log} \frac{ M_S^2}{m_t^2}
\end{flalign}

\DUrole{raw-tex}{\noindent}
where the Fermi constant \DUrole{raw-tex}{$G_F\sim 10^{-5}\,\, GeV^-{2}$} was introduced using its relation to the
v.e.v.s.    \DUrole{raw-tex}{$v=(\sqrt{2}G_F)^{-1/2}$}. The contribution \DUrole{raw-tex}{$(\ref{eq68})$} increases quartically with
the top quark mass and logarithmically with  the stop mass shifhting the maximal value of \DUrole{raw-tex}{$M_h$}  upto
\DUrole{raw-tex}{$M_h^{max}\sim 135\,\,\, GeV $}.

The calculation of the radiative corrections to the Higgs boson masses
like  the one-loop contribution given by  \DUrole{raw-tex}{$(\ref{eq68})$}
requires the choice of a renormalisation scheme.  Using the Feynman-diagrammatic approach, one
can write out \DUrole{raw-tex}{\cite{hepph9609331v1}} the leading
one-loop radiative corrections to CP-even Higgs bosons in terms of ``On-Shell'' renormalisation
( \textbf{OS}) parameters, such as pole particle masses and and suitably defined mixing anlgles

\begin{flalign}\label{eq69}
   {\cal M}^2 = \left[ \begin{array}{cc} {\cal M}_{11}^2 + \Delta {\cal M}_{11}^2
  & {\cal M}_{12}^2 + \Delta {\cal M}_{12}^2 \\
    {\cal M}_{12}^2 + \Delta {\cal M}_{12}^2
  &  {\cal M}_{22}^2 + \Delta {\cal M}_{22}^2
 \end{array} \right].
\end{flalign}

\DUrole{raw-tex}{\noindent}
The expressions of \DUrole{raw-tex}{$\Delta{\cal M}_{ij}^2$} can be simplifed to the form

\begin{flalign}\label{eq70}
 \Delta {\cal M} \equiv \frac{3 m_t^4}{ 2\pi^2 v^2sin^2\beta}
   log\left(M_S^2/m_t^2\right)
   \left(\begin{array}{cc}
   0 & 0 \\
    0 & 1
    \end{array}\right)\,.
\end{flalign}

\DUrole{raw-tex}{\noindent}
where effects of squark mixing  were neglected. However the MSSM parameters at weak scale can  be
derived from a set of unified parameters at GUT scale via RG evolution. The parameters
become ``running''  quantities expressed in the Minimal Subtraction (\textbf{MS}) or Dimensional Reduction
(\textbf{DR}) schemas which are usually prefered for the calculations of the radiative corrections  in SUSY.
The   \textbf{DR} scheme preserves SUSY at least up to two-loop order. The approximation \DUrole{raw-tex}{$(\ref{eq70})$}
can be improved if someone  replaces the \textbf{OS} parameter \DUrole{raw-tex}{$m_t$} by its ``running'' version
\DUrole{raw-tex}{$m_t(\mu_t)$} where \DUrole{raw-tex}{$\mu_t$} scale is choosen in the appropriate way \DUrole{raw-tex}{$$\mu_t = \sqrt{M_S m_t}. $$}
This RG-improvement and including the squark mixing effects transform the  leading one-loop contribution
\DUrole{raw-tex}{$(\ref{eq70})$} to the following form

\begin{flalign}\label{eq71}
 \Delta{\cal M}\equiv \frac{3 m_t^4(\mu_t)}{ 2\pi^2 v^2sin^2\beta}
   \left[log\left(\frac{M_S^2}{m_t^2(\mu_t)}\right)+\frac{X_t^2}{2M_S^2}\left(1-\frac{X_t^2}{6M_S^2}\right)\right]
   \left(\begin{array}{cc}
   0 & 0 \\
    0 & 1
    \end{array}\right)\,.
\end{flalign}

\DUrole{raw-tex}{\noindent}
which successfully reproduces the most important aspects of leading-logarithmic  effects up to the two-loop
order \DUrole{raw-tex}{\cite{hepph9609331v1}}.     The \DUrole{raw-tex}{$X_t$} mixing parameter of the stops is given by
\DUrole{raw-tex}{$$ X_t = A_t - \mu cot\beta, $$}
where the trilinear coupling \DUrole{raw-tex}{$A_t$} between top squatk and the Higgs boson ,
the higgsino mass parameter \DUrole{raw-tex}{$\mu$} are  defined by  the  soft-SUSY breaking scalar potential.
The SUSY scale \DUrole{raw-tex}{$M_S$} is the arithmetic average of the stop masses, ie. \DUrole{raw-tex}{$M_S=\frac{1}{2}(m_{\tilde{t_1}}+m_{\tilde{t_2}})$}
if there are only two  generation families. The Figure \DUrole{raw-tex}{$\ref{fig12}$} illustrates the comparison of
exact one-loop order   \textbf{OS} correction (a dashed  line) with the
approximation \DUrole{raw-tex}{$(\ref{eq69})$}  (a higher dotted line)
and    RG-improved  one \DUrole{raw-tex}{$(\ref{eq70})$}  (a lower dotted line). The largest difference between the approaches
is  \DUrole{raw-tex}{$\sim 3\%$} at large values of the scale \DUrole{raw-tex}{$\mu_t$}..
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/hhhfig1_rotate90.pdf}}
\caption{The radiatively corrected light CP-even Higgs mass is plotted
as a function of \DUrole{raw-tex}{$M_S$} for \DUrole{raw-tex}{$\tan\beta=20$} and \DUrole{raw-tex}{$M_A= 1$} TeV.
The one-loop leading logarithmic computation {[}dashed line{]}
is compared with the leading \DUrole{raw-tex}{$(\ref{eq69})$} approximation of
{[}higher dotted line{]}, and its RG-improvement  \DUrole{raw-tex}{$(\ref{eq70})$}
{[}lower dotted line{]}  \DUrole{raw-tex}{\label{fig12}}}
\end{figure}

In the approximation \DUrole{raw-tex}{\cite{hepph9609331v1}}, the one-loop corrected CP-even Higgs boson masses
are obtained
by diagonalizing the mass matrix \DUrole{raw-tex}{$(\ref{eq69})$}. The masses are simply defined then
\DUrole{raw-tex}{\cite{37 hepph05031173v2}}, \DUrole{raw-tex}{\cite{hepph05031173v2}}    as

\begin{flalign}\label{eq71}
 M_{h,H}^2 = \frac{1}{2} (M_A^2+ M_Z^2+\Delta {\cal M}_{22}) \left[ 1 \mp
 \sqrt{1- 4 \frac{ M_Z^2 M_A^2 \cos^2 2\beta +\Delta {\cal M}_{22}( M_A^2 \sin^2\beta +
 M_Z^2 \cos^2\beta)} {(M_A^2+ M_Z^2+\Delta {\cal M}_{22})^2} } \right]. \
\end{flalign}

\DUrole{raw-tex}{\noindent}
The upper bound on \DUrole{raw-tex}{$M_h$} reaches its maximum at \DUrole{raw-tex}{$M_A\gg M_Z$} for a givent \DUrole{raw-tex}{$tan \beta$}
in accordance to

 \begin{flalign}\label{eq71}
  M_{h}^{max}={M_Z}^2 \cos^22\beta + \Delta {\cal M}_{22} \sin\beta  + \frac{M_Z^4}{M_A^2}\left(cos^22\beta-1\right) + \nonumber \\
  \frac{\Delta {\cal M}_{22}}{M_A^2}\left( M_Z^2(\cos\beta-cos^22\beta+\cos 4\beta\sin\beta) + \sin\beta(\sin\beta-1)\Delta {\cal M}_{22}\right).
\end{flalign}

\DUrole{raw-tex}{\noindent}
The impact of the radiative correction \DUrole{raw-tex}{$(\ref{eq69})$} is shown in the Figure \DUrole{raw-tex}{$\ref{fig13}$}.  Even at small values
of \DUrole{raw-tex}{$\tan\beta$}, the corrected   \DUrole{raw-tex}{$M_h$}  reaches \DUrole{raw-tex}{$~130 $} GeV
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Mh_RC_Xt0_tanb5.png}}
\caption{The tree-level and radiatively corrected light CP-even Higgs mass as  functions of \DUrole{raw-tex}{$M_A$}
are plotted for \DUrole{raw-tex}{$tan\beta=5,\,\, M_S=1\,TeV,\,\,m_t=175,\,GeV  $}  \DUrole{raw-tex}{\label{fig13}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The impact of the radiative correction is shown in the Figure \DUrole{raw-tex}{$\ref{fig13}$}.  Even at small values
of \DUrole{raw-tex}{$\tan\beta$}, the corrected   \DUrole{raw-tex}{$M_h$}  reaches \DUrole{raw-tex}{$~130 $} GeV
The effects of stop mixing in \DUrole{raw-tex}{$(\ref{eq70})$} brings   the corrected \DUrole{raw-tex}{$M_h^{max}$} to  the values
up-to 3-4 GeV higher than  the  \DUrole{raw-tex}{$X_t=0$} case in the Figure \DUrole{raw-tex}{$\ref{fig13}$}.
The upper bound on \DUrole{raw-tex}{$M_h$} in the  ``maximal mixing'' scenario: \DUrole{raw-tex}{$$X_t=A_t-\mu cot\beta \sim  2M_S$$}  is
illustrated by the Figure \DUrole{raw-tex}{$\ref{fig14}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Mh_RC_nomix_maxmix_tanb5.png}}
\caption{The radiatively corrected light CP-even Higgs mass as  functions of \DUrole{raw-tex}{$M_A$} for \DUrole{raw-tex}{$X_t\sim 2M_S$} and
\DUrole{raw-tex}{$X_t=0$} are plotted at \DUrole{raw-tex}{$tan\beta=5,\,\, M_S=1\,TeV,\,\,m_t=175,\,GeV  $}  \DUrole{raw-tex}{\label{fig14}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The \DUrole{raw-tex}{$M_S$} SUSY scale plays the crucial role as well. The radiative corrections are
enchanced when the logarithm of \DUrole{raw-tex}{$(\ref{eq69})$}  is
large that happens at large values of \DUrole{raw-tex}{$M_S$} corresponding
to heavy stops. Inreasing \DUrole{raw-tex}{$M_S$} up to 2 TeV brings the mass \DUrole{raw-tex}{$M_h$} to be 135 GeV even in the ``no-mixing'' scenario.
The Figure \DUrole{raw-tex}{$\ref{fig15}$} shows how the mass of the CP-even lightest Higgs boson  depends on \DUrole{raw-tex}{$X_t$} at
different values of \DUrole{raw-tex}{$M_S$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Mh_RC_MS_tanb10.png}}
\caption{The radiatively corrected light CP-even Higgs mass as  functions of \DUrole{raw-tex}{$X_t$} at  \DUrole{raw-tex}{$tan\beta=5,\,\, M_S=1(2)\,TeV,\,\,m_t=175,\,GeV  $}  \DUrole{raw-tex}{\label{fig15}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The impact of \DUrole{raw-tex}{$tan \beta$}  parameter on \DUrole{raw-tex}{$M_h$}  is depicted in  the Figure \DUrole{raw-tex}{$\ref{fig16}$}. The maximal
value of \DUrole{raw-tex}{$M_h$} as \DUrole{raw-tex}{$\sim 135\,$}GeV is reached when \DUrole{raw-tex}{$tan \beta$}  increases to the high values.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.900\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Mh_Mtop_tanb10.png}}
\caption{The radiatively corrected light CP-even Higgs mass as  functions of \DUrole{raw-tex}{$tan \beta$} at
\DUrole{raw-tex}{$M_S=1\,TeV,\,\,m_t=175,\,GeV,\,\,\,X_t=2M_S(0)  $}  \DUrole{raw-tex}{\label{fig16}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The green band in the Figure \DUrole{raw-tex}{$\ref{fig16}$} is an impact on \DUrole{raw-tex}{$M_h$} of the uncertainty from the
top quark mass \DUrole{raw-tex}{$m_{top}$}  assumed to be \DUrole{raw-tex}{$\pm 5\,$}GeV.


\subsection{3.6~~~The implication of SM Higgs boson for MSSM%
  \label{the-implication-of-sm-higgs-boson-for-mssm}%
}

The Higgs-like particle with a mass of about \DUrole{raw-tex}{$125 GeV$} has been discovered at the LHC \DUrole{raw-tex}{\cite{1 hepph1302.7033}},
\DUrole{raw-tex}{\cite{2 hepph1302.7033v1}}.This new state is compatible with predictions of SM.
In the minimal SUSY extensiom of the Standard Model, MSSM, it is obviously to interpret the new state
as the light CP-even Higgs boson \DUrole{raw-tex}{$h$} \DUrole{raw-tex}{\cite{8-11 hepph1302.7033v1}}

As it was pointed in the previous  section \DUrole{raw-tex}{$\ref{radiative-corrections-and-the-upper-bound-on-m-h-label-secrcmh}$}
out, the radiative effects push \DUrole{raw-tex}{$M_h$} upward from
the tree-level bound \DUrole{raw-tex}{$M_Z$} to the mass range \DUrole{raw-tex}{$123\,\, GeV \leq M_h\leq 135\,\, GeV$} which includes
the mass of SM Higgs boson \DUrole{raw-tex}{$M_{H_SM}\simeq 125\,\, GeV$}.At high  \DUrole{raw-tex}{$M_A$} values, \DUrole{raw-tex}{$M_A \gg M_Z$},
the Higgs sector of MSSM turns into the so-called decoupling regime in which the CP-even neutral light scalar
state  \DUrole{raw-tex}{$h$} has almost exactly  the  properties  of the SM Higgs particle like its couplings to fermions
and gauge bosons.

Indeed the Yukawa Lagrangian given by \DUrole{raw-tex}{$(\ref{eq52})$}  and \DUrole{raw-tex}{$(\ref{eq53})$}  in terms of the field content
(of the first fermion family) can be written as

 \begin{flalign}\label{eq72}
  {\cal L}_{\rm Yuk}=  - \lambda_u [ \bar u P_L u H_2^0  - \bar u P_L d H_2^+ ] - \lambda_d [ \bar d P_L d H_1^0  - \bar d P_L u H_1^- ] + {\rm h.c.},
\end{flalign}

\DUrole{raw-tex}{\noindent}
where  \DUrole{raw-tex}{$P_{L(R)}$} are left- and right-handed projection operators. The 3x3 flavor mixing matrices \DUrole{raw-tex}{$y^{U,D,L}_{ab}$}
introduced in \DUrole{raw-tex}{$(\ref{eq53})$}  are assumed to be diagonal having the elements \DUrole{raw-tex}{$\lambda_i$} with \DUrole{raw-tex}{$i=u,d\,\, etc$}.
The fermion masses are generated after EWSB and they are related to Yukawa coupling \DUrole{raw-tex}{$\lambda_i$}  by

  \begin{flalign}\label{eq73}
  \lambda_u= \frac{ \sqrt{2} m_u} {v_2} = \frac{ \sqrt{2} m_u} {v \sin\beta},\,\,\, \lambda_d= \frac{ \sqrt{2} m_d} {v_1} = \frac{ \sqrt{2} m_d} {v \cos\beta}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Expressing the fields \DUrole{raw-tex}{$H_1$}  and \DUrole{raw-tex}{$H_2$}  in terms of the physical fields \DUrole{raw-tex}{$H,h$}  provided by
\DUrole{raw-tex}{$(\ref{eq65})$} and \DUrole{raw-tex}{$(\ref{eq66})$}, the Yukawa Lagrangian takes  the following form

  \begin{flalign}\label{eq74}
  {\cal L}_{\rm Yuk}= -\frac{g_2 m_u}{2M_W \sin \beta} \left[\bar{u}u (H\sin
  \alpha+ h\cos \alpha) - i \bar{u} \gamma_5u \, A \cos\beta \right] \nonumber \\
  -\frac{g_2 m_d}{2M_W \cos \beta} \left[\bar{d}d (H\cos \alpha -
   h\sin \alpha) - i \bar{d} \gamma_5 d \, A \sin\beta \right] \nonumber \\
   +\frac{g_2}{2\sqrt{2}M_W} V_{ud} \, \left\{ H^+ \bar{u} [m_d \tb (1+\gamma_5)
   + m_u{\rm cot}\beta (1-\gamma_5)] d + {\rm h.c.} \right\},
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$V_{ud}$}  is the element of CKM  matrix. As a cosequence of \DUrole{raw-tex}{$(\ref{eq74})$},
the couplings of neutral Higgs bosons \DUrole{raw-tex}{A,H,h}
to fermions strongly depend on  the angle of the rotation \DUrole{raw-tex}{$\alpha$} and the parameter \DUrole{raw-tex}{$tan\beta$}.
The next Table \DUrole{raw-tex}{$\ref{tbl3}$} summarizes them relative to SM values.

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.203\DUtablewidth}|p{0.331\DUtablewidth}|p{0.307\DUtablewidth}|}
\caption{Neutral Higgs bosons couplings to the fermions \DUrole{raw-tex}{\label{tbl3}}}\\
\hline
\textbf{%
Higgs
} & \textbf{%
\DUrole{raw-tex}{$(\Phi\bar{u}u)$}
} & \textbf{%
\DUrole{raw-tex}{$(\Phi\bar{d}d)$}
} \\
\hline
\endfirsthead
\caption[]{Neutral Higgs bosons couplings to the fermions \DUrole{raw-tex}{\label{tbl3}} (... continued)}\\
\hline
\textbf{%
Higgs
} & \textbf{%
\DUrole{raw-tex}{$(\Phi\bar{u}u)$}
} & \textbf{%
\DUrole{raw-tex}{$(\Phi\bar{d}d)$}
} \\
\hline
\endhead
\multicolumn{3}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

\DUrole{raw-tex}{$\bf H_{SM}$}
 & 
1
 & 
1
 \\
\hline

\DUrole{raw-tex}{$\bf H$}
 & 
\DUrole{raw-tex}{$cos \alpha/sin \beta$}
 & 
\DUrole{raw-tex}{$-sin \alpha/cos\beta$}
 \\
\hline

\DUrole{raw-tex}{$\bf h$}
 & 
\DUrole{raw-tex}{$sin \alpha/sin \beta$}
 & 
\DUrole{raw-tex}{$cos \alpha/cos\beta$}
 \\
\hline

\DUrole{raw-tex}{$\bf A$}
 & 
\DUrole{raw-tex}{$cot \beta$}
 & 
\DUrole{raw-tex}{$tan \beta$}
 \\
\hline
\end{longtable}

\DUrole{raw-tex}{\noindent}
The neutral Higgs boson couplings to the gauge bosons \DUrole{raw-tex}{$V=W^{\pm},Z,\gamma$} are ruled out from the kinetic term of
the SUSY Lagrangian  \DUrole{raw-tex}{${\cal L}_{guage}$} taken from the equation \DUrole{raw-tex}{$(\ref{eq52})$}.
The following Table \DUrole{raw-tex}{$\ref{tbl4}$}  contains the definitions of \texttt{one-scalar-two-gauge-boson},
\DUrole{raw-tex}{$(\Phi VV)$},
couplings as well as the  couplings of \texttt{two-scalars-one-gauge-bosons},  \DUrole{raw-tex}{$(\Phi AV)$},  interaction.

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.141\DUtablewidth}|p{0.438\DUtablewidth}|p{0.366\DUtablewidth}|}
\caption{Neutral Higgs bosons couplings to the gauge bosons  \DUrole{raw-tex}{\label{tbl4}}}\\
\hline
\textbf{%
Higgs
} & \textbf{%
\DUrole{raw-tex}{$(\Phi ZZ),(\Phi W^+W^-)$}
} & \textbf{%
\DUrole{raw-tex}{$(\Phi AZ)$}
} \\
\hline
\endfirsthead
\caption[]{Neutral Higgs bosons couplings to the gauge bosons  \DUrole{raw-tex}{\label{tbl4}} (... continued)}\\
\hline
\textbf{%
Higgs
} & \textbf{%
\DUrole{raw-tex}{$(\Phi ZZ),(\Phi W^+W^-)$}
} & \textbf{%
\DUrole{raw-tex}{$(\Phi AZ)$}
} \\
\hline
\endhead
\multicolumn{3}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

\DUrole{raw-tex}{$\bf H_{SM}$}
 & 
\DUrole{raw-tex}{$ig_ZM_Z, ig_WM_W$}
 & 
0
 \\
\hline

\DUrole{raw-tex}{$\bf H$}
 & 
\DUrole{raw-tex}{$ig_ZM_Zcos(\beta-\alpha),ig_WM_Wcos(\beta-\alpha)$}
 & 
\DUrole{raw-tex}{$g_Zcos(\beta-\alpha)(p_2^\mu+p_3^\mu)$}
 \\
\hline

\DUrole{raw-tex}{$\bf h$}
 & 
\DUrole{raw-tex}{$ig_ZM_Zsin(\beta-\alpha),ig_WM_Wsin(\beta-\alpha)$}
 & 
\DUrole{raw-tex}{$-g_Zsin(\beta-\alpha)(p_2^\mu+p_3^\mu)$}
 \\
\hline

\DUrole{raw-tex}{$\bf A$}
 & 
0,0(CP-invariance forbids)
 & 
0(CP-invariance forbids)
 \\
\hline
\end{longtable}

\DUrole{raw-tex}{\noindent}
where the couplings \DUrole{raw-tex}{$g_Z$} \DUrole{raw-tex}{$g_W$} were defined  accordingly
\DUrole{raw-tex}{$$g_Z=g_2/cos \theta_W,\,\,\, g_W=g_2.$$}   The Table \DUrole{raw-tex}{\ref{tbl4}}  doesn't contains the couplings like
\DUrole{raw-tex}{$(\Phi\gamma\gamma), $(\Phi Z\gamma)$}, beause it is kinematically forbidden.

The difference between angles \DUrole{raw-tex}{$\alpha$} and \DUrole{raw-tex}{$\beta$} determines the strength of the \texttt{scalar-graug-bosons}
interactions. The squared sums of the couplings given by Table \DUrole{raw-tex}{$\ref{tbl4}$} satisfy to the rules of complement

  \begin{flalign}\label{eq75}
   (hVV)^2 + (HVV)^2 = (H_{SM}VV)^2, \nonumber \\
   (hAZ)^2 + (HAZ)^2 = (H_{SM}ZZ)^2/(4M_Z^2)
\end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can rewrite the expressions  like \DUrole{raw-tex}{$cos \alpha/sin \beta$}   introduced in  the Table \DUrole{raw-tex}{$\ref{tbl3}$}
in terms of \DUrole{raw-tex}{$sin (\beta-\alpha)$}   and \DUrole{raw-tex}{$\tan\beta$}

  \begin{flalign}\label{eq76}
   (hdd) = -\frac{sin \alpha}{cos\beta}  = sin(\beta-\alpha) - tan\beta cos(\beta-\alpha), \nonumber \\
   (huu) = \frac{cos \alpha}{sin\beta}  = sin(\beta-\alpha) + cot\beta cos(\beta-\alpha), \nonumber \\
   (Hdd) = \frac{cos \alpha}{cos\beta}  = cos(\beta-\alpha) + tan\beta sin(\beta-\alpha), \nonumber \\
   (Huu) = \frac{sin \alpha}{sin\beta}  = cos(\beta-\alpha) - cot\beta sin(\beta-\alpha)
\end{flalign}

\DUrole{raw-tex}{\noindent}
Recalling the rotation matrix \DUrole{raw-tex}{$Z^H$} given by \DUrole{raw-tex}{$(\ref{eq64})$} and \DUrole{raw-tex}{$(\ref{eq65})$},  the parameter of
rotation  matrix \DUrole{raw-tex}{$\alpha$} is not free and it is
determined by \DUrole{raw-tex}{$\tan\beta$}  and the mass eigenbasis \DUrole{raw-tex}{$M_A,M_h,M_H$}

\begin{flalign}\label{eq76}
\cos 2\alpha = -\cos2\beta \, \frac{M_A^2 - M_Z^2}{ M_H^2-M_h^2} \ , \
\sin 2\alpha = -\sin2\beta \, \frac{M_H^2 + M_h^2}{ M_H^2-M_h^2},\nonumber \\
\end{flalign}

\DUrole{raw-tex}{\noindent}
Using \DUrole{raw-tex}{$(\ref{eq76})$}, it is easy to show that

\begin{flalign}\label{eq77}
cos^2(\beta-\alpha) = \frac{M_h^2(M_Z^2 - M_h^2)}{M_A^2(M_H^2-M_h^2)}.
\end{flalign}

\DUrole{raw-tex}{\noindent}
As it follows from
the Tables \DUrole{raw-tex}{$\ref{tbl3}$}, \DUrole{raw-tex}{$\ref{tbl4}$} and the expression \DUrole{raw-tex}{$(\ref{eq77})$},
the decoupling regime \DUrole{raw-tex}{$M_A\gg M_Z\sim M_h$} leads to \DUrole{raw-tex}{$$cos(\beta-\alpha)\rightarrow 0, $$} what
means the suppressions of the \DUrole{raw-tex}{$H-$}type couplings to gauge bosons
and the enhancement of \DUrole{raw-tex}{$h-$}type  couplings to gauge bosons up to values expected from SM
\DUrole{raw-tex}{$$ (h\bar{u}u,h\bar{d}d,hZZ,hW^+W^-)=(H_{SM}\bar{u}u,H_{SM}\bar{d}d,H_{SM}ZZ,H_{SM}W^+W^-)$$}
\DUrole{raw-tex}{$$ (H\bar{u}u)=-cot\beta(H_{SM}\bar{u}u),\,\, (H\bar{d}d)=tan\beta(H_{SM}\bar{d}d),\,\,\,
(HZZ,HW^+W^-)=0$$}

At large values of  \DUrole{raw-tex}{$M_A$}, neutral CP-even \DUrole{raw-tex}{$H$} and CP-odd \DUrole{raw-tex}{$A$} Higgs bosons degenerate in mass
\DUrole{raw-tex}{$M_A\simeq M_H$} and their couplings are \DUrole{raw-tex}{$tan\beta$} enhanced (suppressed)
for down-type (up-type) quarks. The \DUrole{raw-tex}{$h-$}couplings reach their SM values more quickly if \DUrole{raw-tex}{$tan\beta$} is
large as one can see  in the Figures \DUrole{raw-tex}{$\ref{fig17}$} and \DUrole{raw-tex}{$\ref{fig18}$} of \DUrole{raw-tex}{\cite{hep-ph0503173}}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/ghVV.PNG}}
\caption{The normalized couplings squared of the CP-{}-even MSSM
neutral Higgs boson \DUrole{raw-tex}{$h$} to gauge bosons as a function of \DUrole{raw-tex}{$M_A$} for
two values \DUrole{raw-tex}{$tan\beta=3$ and 30, in the no mixing (cyan and magenta lines) and maximal mixing
(blue and red lines) scenarios. The radiative correction is included.  `\label{fig17}}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/ghuu.PNG}}
\caption{The normalized couplings squared of the CP-{}-even MSSM
neutral Higgs boson \DUrole{raw-tex}{$h$} to fermions as a function of \DUrole{raw-tex}{$M_A$} for
two values \DUrole{raw-tex}{$tan\beta=3$} and 30, in the no mixing (cyan and magenta lines) and maximal mixing
(blue and red lines) scenarios. The radiative correction is included.  \DUrole{raw-tex}{\label{fig18}}}
\end{figure}


\subsection{3.7~~~Benchmark scenarios in constrained and phenomenological MSSM%
  \label{benchmark-scenarios-in-constrained-and-phenomenological-mssm}%
}

The detailed phenomenological analysis of the masses and couplings of the SUSY particles in  MSSM
and the comparison of them with the results and expectations of the present and future experiments are
extremely difficult tasks because of the large number of parameters. The most general soft-SUSY breaking
Lagrangian of MSSM  may contain more than 100 free parameters and this number is restricted to 22 if one
imposes some phenomenological constraints. Such phenomenological MSSM (\textbf{pMSSM}) \DUrole{raw-tex}{\cite{7 hepph1123028}}
\DUrole{raw-tex}{\cite{2 hepph0406166v3}}
implies the CP conservation,the diagonal sfermion mass matrices for all flavors, coupling matrices universality
for \DUrole{raw-tex}{$1^st$} and \DUrole{raw-tex}{$2^nd$} fermion generations. The model involves the parameters in additions to those of
SM:
%
\begin{itemize}

\item \DUrole{raw-tex}{$tan\beta$}

\item the soft-SUSY breaking  Higgs parameters \DUrole{raw-tex}{$M_{H_1}^2$} and   \DUrole{raw-tex}{$M_{H_2}^2$}  (or \DUrole{raw-tex}{$M_A$} and \DUrole{raw-tex}{$\mu$} at EWSB )

\item the gaugino mass parameters \DUrole{raw-tex}{$M_1$}, \DUrole{raw-tex}{$M_2$}, \DUrole{raw-tex}{$M_3$}

\item the diagonal sfermion mass parameters \DUrole{raw-tex}{$m_{\tilde{f_{L,R}}}$}: five for each fermion generation

\item the soft-breaking sfermion couplings \DUrole{raw-tex}{$A_f$}: three for each fermion family

\end{itemize}

However there are well motivated theoretical models where the soft-SUSY breaking parameters obey
a number of universal boundary conditions at some very high scale, such as GUT scale \DUrole{raw-tex}{$M_{GUT}\sim 10^{15}$}GeV.
This reduce the number of the  parameters to \DUrole{raw-tex}{$4 \div 6$}  fundamental parameters in MSSM called
contstrained MSSM (\textbf{cMSSM}). Such models differ
among themselves with respect to the assumptions made regarding the SUSY breaking mechanisms. This is
the case of the models of gravity mediated \DUrole{raw-tex}{\cite{3 hepph0406166v3}}, \textbf{mSUGRA}, gauge mediated
\DUrole{raw-tex}{\cite{4 hepph04061166v3}},(\textbf{GMSB}), or anomaly mediated \DUrole{raw-tex}{\cite{5 hepph04061166v3}}, (\textbf{AMSB}),
SUSY breaking modes. The universality hypothesis leaves only the following parameters in \textbf{cMSSM}
%
\begin{itemize}

\item the higgsino mass soft-SUSY breaking parameter, \DUrole{raw-tex}{$\mu$}

\item the universal mass  soft-SUSY breaking parameter for  all scalars, \DUrole{raw-tex}{$m_0$}

\item the universal mass  soft-SUSY breaking parameter for  all fermions, \DUrole{raw-tex}{$m_{1/2}$}

\item the universal soft-SUSY breaking trilinear coupling, \DUrole{raw-tex}{$A_0$}

\item \DUrole{raw-tex}{$\tan\beta$}

\end{itemize}

The parameters in \textbf{cMSSM} and \textbf{pMSSM} at the weak scale are derived from the considered  parameters
at the GUT scale via \textbf{RG} evolution which is usually calculated in \textbf{DR} scheme adopted to SUSY
\DUrole{raw-tex}{\cite{16 hepph0406166v3}}. Also the high-scale boundary conditions are applied to gauge couplings \DUrole{raw-tex}{$g_1,g_2$}
or gaugino  mass parameters \DUrole{raw-tex}{$M_1$} and \DUrole{raw-tex}{$M_2$}

\begin{flalign}\label{eq78}
g_2(M_{GUT})=\sqrt{\frac{5}{3}}g_1(M_{GUT}),\nonumber \\
M_1(M_{GUT})=\frac{5}{3}\frac{\sin\theta_W}{\cos\theta_{W}}M_2(M_{GUT}),
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$g_2(M_2)$} and \DUrole{raw-tex}{$g_1(M_1)$} are  associated with the \DUrole{raw-tex}{$SU(2)_L$}  and \DUrole{raw-tex}{$U(1)_Y$} gauge group of the
MSSM, respectively. Afterwards EWSB constraints are imposed at the weak scale to exclude regions in the
MSSM parameter space which are not compatible (usually with 95\% Confidence Level \textbf{CL}) with
experimental data or violates theoretical assumptions like stability of  \textbf{LSP}
or Yukawa couplings unification \DUrole{raw-tex}{\cite{43 GUT-KAZAKOV}}. These constraints have been discussed in details
in \DUrole{raw-tex}{\cite{33 hepph1207.1348v2}}.

I will illustrate the MSSM mass spectrum  and the parameter space in the frameworks of some scenarios applied  to mSUGRA (cMSSM) and pMSSM models. In the case of mSUGRA, the
Snowmass Points and Slopes \DUrole{raw-tex}{\cite{34 hepph0406166v3}} benchmark scenario is  one of the most
recommended as an exclusive and sufficient collection of points and  continuous sets of the parameters
for studies of SUSY phenomenology. \textbf{SPS1a}  is a \texttt{standard}  mSUGRA point of intermediate \DUrole{raw-tex}{$\tan\beta=10$}
and with the non-negligible mixing bewteen staus \DUrole{raw-tex}{$(\tilde{\tau}_1,\tilde{\tau}_2)$} or sbottoms
\DUrole{raw-tex}{$(\tilde{b}_1,\tilde{b}_2)$}
leading to a significant mass splitting between
them. Thefore neutralinos and charginos  decay predominantly into staus and taus.
The sparticle spectra of the SPS1a point is shown in the Figure \DUrole{raw-tex}{$\ref{fig19}$}.
The SPS1a point is defined as
\DUrole{raw-tex}{$$ SPS1a:\,\,\, m_0=100\,\,GeV,\,\, m_{1/2}=250\,\,GeV,\,\, A_0=-100\,\,GeV,\,\, \tan\beta=10,\,\, \mu>0$$}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/SPS1a.png}}
\caption{The SUSY particle spectra and decays for the benchmark points corresponding to SPS1a \DUrole{raw-tex}{\label{fig19}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The mass upper bounds on \DUrole{raw-tex}{$M_h$} can be found using a flat scan of the cMSSM parameters. The task is quite
complicated because the SUSY parameters entering the radiative correction to \DUrole{raw-tex}{$M_h$} are not all independent.
This happens due to their relations in the system of \textbf{RG}  equations.  In addition, the EWSB
constraints must be fullfilled that provides the additional dependence between input parameters.
In order to find the functional dependence of \DUrole{raw-tex}{$M_h$} on the mSUGRA parameters, the scan .
\DUrole{raw-tex}{$$ |A_0|<=5\,\, TeV,\,\, 1\leq \tan\beta \leq 60 $$}
is done \DUrole{raw-tex}{\cite{hepph0406166}} at the fixed parameters  \DUrole{raw-tex}{$m_0=m_{1/2}=1\,TeV$}. The SUSY scale \DUrole{raw-tex}{$M_S$} was floating parameter
which values were functional dependent on \DUrole{raw-tex}{$A_0$} due to the reasons mentioned above.
The Figure \DUrole{raw-tex}{$\ref{fig20}$}  shows   \DUrole{raw-tex}{$M_h$} as the density in the  \DUrole{raw-tex}{$A_0-\tan\beta$}    plane.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/sugra_scan.png}}
\caption{The lighter Higgs boson mass \DUrole{raw-tex}{$M_h$} in the \DUrole{raw-tex}{$\tan\beta-A_0$} plane of the
mSUGRA scenario. Other input parameters are \DUrole{raw-tex}{$m_0 = m_{1/2} = 1$} TeV and
\DUrole{raw-tex}{$\mu>0$}. The SM input parameters are fixed to their default
values. The lighter Higgs boson mass is shown as background density, as
measured by the bar on the right.  Dashed contours are lines of equal
\DUrole{raw-tex}{$M_S$} , dotted contours are lines of equal \DUrole{raw-tex}{$M_H$}. The white region
contains scalars with negative squared mass.    \DUrole{raw-tex}{\label{fig20}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Large stop mixing at \DUrole{raw-tex}{$A_0\sim\, -4\,\,TeV$} and the large radiative correction, as a result, enforce
the mass of the  lightest nuetral CP-even Higgs boson, \DUrole{raw-tex}{$M_h$}  to reach the values \DUrole{raw-tex}{$\sim 128.5\,\,GeV$}
\DUrole{raw-tex}{\cite{hepph0406166v3}}.
The region of the maximal \DUrole{raw-tex}{$M_h$} corresponds to the intermediate  range of \DUrole{raw-tex}{$\tan\beta$} \DUrole{raw-tex}{$$ 10\leq \tan\beta \leq 30.$$}

pMSSM has more predictability in the comparison with cMSSM and it offers
a convinient framework for phenomenological studies.
In general, only a small subset of the parameters appears when looking at the Higgs sector of the pMSSM.
The mixing scenarios defined by the impact of the radiative correction \DUrole{raw-tex}{$(\ref{eq70})$}  on \DUrole{raw-tex}{$M_h$}
\DUrole{raw-tex}{$(\ref{eq71})$}
%
\begin{itemize}

\item maximal mixing: \DUrole{raw-tex}{$X_t=\sqrt{6}M_S$} \DUrole{raw-tex}{$\label{mhmax}$}  (large radiative effects maximise \DUrole{raw-tex}{$M_h$})

\item no-mixing: \DUrole{raw-tex}{$X_t=0$}                 (the much smaller effects reduce the upper bound on \DUrole{raw-tex}{$M_h$})

\end{itemize}

\DUrole{raw-tex}{\noindent}
are often used as benchmarks for the analysis of SUSY within pMSSM \DUrole{raw-tex}{\cite{13 hepph1207.1348}} .
Also the various regimes of the pMSSM Higgs sector \DUrole{raw-tex}{\cite{9 hepph12071348v2}}  can be used as very useful
tools for phenomenological studies while the benchmark scenarios probe the confrontation of MSSM to
the latest LHC and Tevatron data. The theoretists usualy define the regimes of pMSSM in the \DUrole{raw-tex}{$(\tan\beta,M_A)$}
plane  in accordance to
%
\begin{itemize}

\item decoupling regime: \DUrole{raw-tex}{$\cos^2(\beta-\alpha) \leq 0.05$}
%
\begin{itemize}

\item the region in \DUrole{raw-tex}{$(M_A,\tan\beta)$}:  \DUrole{raw-tex}{$M_A \gesim 300\,\,GeV$} for low \DUrole{raw-tex}{$\tan\beta\lsim 10$}
and   \DUrole{raw-tex}{$M_A \sim M_h^{max}$} for intermidiate and high \DUrole{raw-tex}{$\tan\beta\gesim 10$}

\item the CP-{}-even \DUrole{raw-tex}{$h$} boson reaches its maximal mass value \DUrole{raw-tex}{$M_h^{\rm max}$} and its
couplings to fermions and gauge bosons (as well as its self-{}-coupling) become
SM-{}-like \DUrole{raw-tex}{\cite{21 hepph12071348v2}}.

\end{itemize}

\item anti-decoupling regime: \DUrole{raw-tex}{$\cos^2(\beta-\alpha) \geq 0.95$}
%
\begin{itemize}

\item the region in \DUrole{raw-tex}{$(M_A,\tan\beta)$}:  \DUrole{raw-tex}{$M_A \lesim M_h^{\rm max}$} for all \DUrole{raw-tex}{$\tan\beta$}

\item this is  exactly opposite to the decoupling regime.
The roles of the \DUrole{raw-tex}{$h$} and \DUrole{raw-tex}{$H$} bosons are reversed. \DUrole{raw-tex}{\cite{22 hepph12071348v2 }}

\end{itemize}

\item intense-{}-coupling regime: \DUrole{raw-tex}{$0.05 \leq \cos^2(\beta-\alpha) \leq 0.7\,\,\,\tan\beta \leq 10 $}
%
\begin{itemize}

\item the region in \DUrole{raw-tex}{$(M_A,\tan\beta)$}: \DUrole{raw-tex}{$M_A\simM_h^{max}$}

\item the three neutral Higgs bosons \DUrole{raw-tex}{$h,H$} and \DUrole{raw-tex}{$A$} (as well as the charged Higgs
particles) have comparable masses, \DUrole{raw-tex}{$M_h \sim M_H \sim M_A \sim M_h^{\rm max}$} .
The mass degeneracy is more effective when \DUrole{raw-tex}{$\tan\beta$} is large.   \DUrole{raw-tex}{\cite{23 hepph12071348v2 }}

\end{itemize}

\item intermediate-{}-coupling regime:  \DUrole{raw-tex}{$M_A\leqsim 140\,\, GeV$} and  \DUrole{raw-tex}{$g_{hbb}^2~{\rm and}~g_{Hbb}^2 \geq 50$}
%
\begin{itemize}

\item the region in \DUrole{raw-tex}{$(M_A,\tan\beta)$}:   \DUrole{raw-tex}{$M_A \lesim 300-500\,\,GeV $} and \DUrole{raw-tex}{$\tan\beta\lesim 5-10$}

\item there is no strongly enchancement or suppresion of the Higgs-fermion couplings that challenges
the LEP data.

\end{itemize}

\end{itemize}

The different regimes are depicted in the Figure \DUrole{raw-tex}{$\ref{fig21}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/maximalmix_MS2_regimes.png}}
\caption{The parameter space for the various regimes of the MSSM Higgs
sector in the \DUrole{raw-tex}{$[M_A,\tan\beta]$} plane. \DUrole{raw-tex}{\label{fig21}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The scan \DUrole{raw-tex}{$$1\leq\tan\beta\greq 50\,\,\, 50\,\, GeV\leq M_A \geq 450\,\, GeV$$}
of pMSSM in the maximal mixing scenario is shown in the Figure \DUrole{raw-tex}{$\ref{fig22}$}.
The implication of the SM Higgs boson significantly constraints the available
parameter space as one can see in the Figure \DUrole{raw-tex}{$\ref{fig22}$}. Following to the Figure \DUrole{raw-tex}{$\ref{fig21}$},
the region of SM-like Higgs boson  corresponds to the decoupling  regime.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/PlotpMSSM_withSM_Higgs.pdf}}
\caption{The parameter space for the maximal mixing of the  pMSSM Higgs
sector in the \DUrole{raw-tex}{$[M_A,\tan\beta]$} plane. \DUrole{raw-tex}{\label{fig22}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The pMSSM spectrum in the maximal mixing scenario is calculated  and illustrated by the
Figure \DUrole{raw-tex}{$\ref{fig23}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/mhmax_spectrum.pdf}}
\caption{The mass spectrum of pMSSM in the maximal mixing scenario. \DUrole{raw-tex}{\label{fig23}}}
\end{figure}


\section{4~~~The neutral MSSM Higgs Boson at LHC%
  \label{the-neutral-mssm-higgs-boson-at-lhc}%
}


\subsection{4.1~~~The production of the MSSM Higgs boson  at LHC%
  \label{the-production-of-the-mssm-higgs-boson-at-lhc}%
}

A Higgs-like particle with a mass around 125-126 GeV has been discovered at the LHC.
CMS experiment analysed the full data sample of proton-proton collisions collected in 2011 and
in 2012 up until June 18. The data of \DUrole{raw-tex}{$5.1\,\, fb^{-1}$`of integrated
luminocity recorded in 2011 at `$\sqrt{S}=7$}TeV and of \DUrole{raw-tex}{$5.3\,\, fb^{-1}$} at \DUrole{raw-tex}{$\sqrt{S}=8$}TeV
in 2012 were processed and reconstructed to claim the discovery of the new particle.

Within the current experimental uncertainties, the new state is compatible with Standard Model Higgs boson
and with the Higgs sector of Minimal Supersymmetric Standard Model. The most important production
mechanisms of the MSSM neutral CP-even Higgs bosons are those that involve gauge bosons and top quarks because of
the Higgs boson couples more likely to the massive particles. In the decoupling limit,
the MSSM scalar sector effectively reduces to the SM features, however the region of large \DUrole{raw-tex}{$tan\beta$},
(the red area in the Figure  \DUrole{raw-tex}{$\ref{fig21}$}) , MSSM Higgs boson couplings to down-type fermions are
strongly enhanced. Thus the bottom quarks will then play a much more important role     than in SM case.

The main processes for the neutral MSSM CP-even Higgs boson production are
%
\begin{itemize}

\item the associated production with \DUrole{raw-tex}{$W/Z$} bosons \DUrole{raw-tex}{\cite{241,242hep-ph0503172v2}}, \DUrole{raw-tex}{$q\bar{q}\rightarrow W/Z+h/H$}

\end{itemize}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.400\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WZ_Higgs_production_Feynman.pdf}}
\caption{Higgs-strahlung: associated \DUrole{raw-tex}{$h/H$} production with \DUrole{raw-tex}{$W/Z$} \DUrole{raw-tex}{\label{fig24}}}
\end{figure}
%
\begin{itemize}

\item vector boson fusion \DUrole{raw-tex}{\cite{112,243,244hep-ph0503172v2}}, \DUrole{raw-tex}{$qq\rightarrow V^{*}V^{*}\rightarrow qq+h/H$}

\end{itemize}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.400\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/VectorFusion_Higgs_production_Feynman.pdf}}
\caption{WW/ZZ fusion production of \DUrole{raw-tex}{$h/H$} \DUrole{raw-tex}{\label{fig25}}}
\end{figure}
%
\begin{itemize}

\item gluon-gluon fusion \DUrole{raw-tex}{\cite{185hep-ph0503172v2}}, \DUrole{raw-tex}{$gg\rightarrow h/H/A$}

\end{itemize}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.400\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/GluonFusion_Higgs_production_Feynman.pdf}}
\caption{gluon-gluon fusion production of \DUrole{raw-tex}{$h/H/A$} \DUrole{raw-tex}{\label{fig26}}}
\end{figure}
%
\begin{itemize}

\item the associated Higgs production with heavy top or bottom quarks

\end{itemize}

\DUrole{raw-tex}{\cite{247,248,249,250hep-ph0503172v2}}, \DUrole{raw-tex}{$gg\rightarrow bb+h/H/A$}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.400\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/basoc_Higgs_production_Feynman.pdf}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.400\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/basoc_Higgs_production_Feynman2.pdf}}
\caption{Higgs boson radiation off bottom quark  \DUrole{raw-tex}{\label{fig27}}}
\end{figure}

The calculation of the Higgs cross sections and decay widths in MSSM can be done in the effective
coupling approximation \DUrole{raw-tex}{\cite{hepph0607308v2}}. The approximation assumes that  the
corresponding SM cross sections treat high-order corrections of SM-type and   SUSY-type on the same
footing. Only rescalings the SM cross sections  with the ratios of the corresponding MSSM
\DUrole{raw-tex}{$g_{HVV}$}, \DUrole{raw-tex}{$g_{Hbb}$} couplings over SM equivalents \DUrole{raw-tex}{\cite{hepph0607308v2}}. The gluon-gluon fusion
process is the dominant production mechanism for SM Higgs  boson at LHC as shown in the Figure \DUrole{raw-tex}{$\ref{fig28}$}
\DUrole{raw-tex}{\cite{LHCXSWG site}}, \DUrole{raw-tex}{\cite{hepph1101.0593v3}}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Higgs_XS_7TeV.png}}
\caption{Standard Model Higgs boson production cross section at \DUrole{raw-tex}{$E_{cm}=7 $}TeV at next-to-next-to-leading order (NNLO)
and at next-to-leading-order (NLO)  \DUrole{raw-tex}{\label{fig28}}}
\end{figure}

The  process, \DUrole{raw-tex}{$gg\rightarrow h/H/A+X$}, dominates in MSSM  at low and moderate \DUrole{raw-tex}{$tan\beta$} values.
The process in MSSM is predominantly mediated not only by top loops as in the SM case,  but bottom and
sbottom loops should be considered involving massive NLO (includfing massive two-loop virtual diagrams) calculations
\DUrole{raw-tex}{\cite{14hepph0304035v2}}.
The increasing importance of the bottom quark loop is a result of the enchancement of the corresponding
Yukawa coupling, \DUrole{raw-tex}{$g^{MSSM}(H/hb\bar{b})$}, as one can see  from  \DUrole{raw-tex}{$(\ref{eq74})$} and the Table \DUrole{raw-tex}{$\ref{tbl3}$}
leads to the following ratios  of the top couplings overt bottom ones at leading order

\begin{flalign}\label{eq79}
 \frac{g^{MSSM}(H,t\bar{t})}{g^{MSSM}(H,b\bar{b})} = -\frac{1}{tan\beta}\frac{m_{t}}{m_b}tan\alpha,\nonumber \\
 \frac{g^{MSSM}(h,t\bar{t})}{g^{MSSM}(h,b\bar{b})} = \frac{1}{tan\beta}\frac{m_{t}}{m_b}\frac{1}{tan\alpha},\nonumber \\
 \frac{g^{MSSM}(A,t\bar{t})}{g^{MSSM}(A,b\bar{b})} = \frac{1}{tan\beta^2}\frac{m_{t}}{m_b}.
\end{flalign}

\DUrole{raw-tex}{\noindent}
The MSSM cross section of the process read  as follows

 \begin{flalign}\label{eq80}
   \sigma(gg\rightarrow h/H/A)^{MSSM} = \frac{g^{2, MSSM}(h/H/A, t\bar{t})}{g^{2,SM}(H, t\bar{t})}
    \sigma^{NNLO,SM}_{t\bar{t}}(gg\rightarrow H) + \nonumber \\
     \frac{g^{2, MSSM}(h/H/A, b\bar{b})}{g^{2,SM}(H, b\bar{b})}\sigma^{NNLO,SM}_{b\bar{b}}(gg\rightarrow H)
    +\nonumber \\
     \frac{g^{MSSM}(h/H/A, t\bar{t})}{g^{SM}(h/H/A, t\bar{t})}\frac{g^{MSSM}(h/H/A, b\bar{b})}
     {g^{SM}(H, b\bar{b})}\sigma^{NNLO,SM}_{t\bar{b}}(gg\rightarrow H),
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$\sigma^{NNLO,SM}_{t\bar{t}}, \sigma^{NNLO,SM}_{b\bar{b}}, \sigma^{NNLO,SM}_{t\bar{b}}$} denote
contributions from top,bottom loops and top-bottom interference, respectevely.  The  NLO
QCD corrections \DUrole{raw-tex}{\cite{203hepph1101.0593v3}}, \DUrole{raw-tex}{\cite{hepph050317v2}}, \DUrole{raw-tex}{\cite{heph9510347}} as well as
the NNLO corrections in the heavy-top-quark limit \DUrole{raw-tex}{\cite{14hepph1101.0593v3}}, \DUrole{raw-tex}{\cite{168hepph1101.0593v3}}
\DUrole{raw-tex}{\cite{hepph0304035}} are used in the terms
\DUrole{raw-tex}{$\sigma^{NNLO,SM}_{t\bar{t}}, \sigma^{NNLO,SM}_{b\bar{b}}, \sigma^{NNLO,SM}_{t\bar{b}}$}
of the equation \DUrole{raw-tex}{$(\ref{eq80})$}.
The Figure \DUrole{raw-tex}{$\ref{fig29}$} shows the cross section \DUrole{raw-tex}{$\sigma^{SM,LO}(gg\rightarrow H)$}
and NLO,  \DUrole{raw-tex}{$\sigma^{SM,NLO}(gg\rightarrow H)$}  and NNLO, \DUrole{raw-tex}{$\sigma^{SM,NNLO}(gg\rightarrow H)$}  QCD
corrections \DUrole{raw-tex}{\cite{hepph0503172v2}} as functions of the Higgs boson mass \DUrole{raw-tex}{$M_H$}  at \DUrole{raw-tex}{$\sqrt{S}=14$}TeV.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/sigsnnlo14.png}}
\caption{The cross sections for \DUrole{raw-tex}{$gg\rightarrow H+X$} fusion mechanism at the LHC (\DUrole{raw-tex}{$E_{cm}=14 $}TeV)
for two factorization and renormalization scales: \DUrole{raw-tex}{$\mu_R=\mu_F=1/2M_H$} (upper curves) and
\DUrole{raw-tex}{$\mu_R=\mu_F=2M_H$} (lower curves). The MRST parton distributions were used \DUrole{raw-tex}{\cite{16,11,12hepph0201206v2}}
\DUrole{raw-tex}{\label{fig29}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The cross section of the gluon-fusion process for the production of the light CP-even
Higgs boson in the MSSM \DUrole{raw-tex}{$m_h^{max}$} scenario is illustrated in the Fugure
\DUrole{raw-tex}{$\ref{fig30}$}. The contributions \DUrole{raw-tex}{$\sigma^{NNLO,SM}_{t\bar{t}}$, $\sigma^{NNLO,SM}_{b\bar{b}}$, $\sigma^{NNLO,SM}_{t\bar{b}}$}
are calculated including  the radiatively corrected  Higgs boson mass given by the equation
\DUrole{raw-tex}{$(\ref{eq71)$}.

The full set of SUSY-QCD corrections \DUrole{raw-tex}{$\mathcal{O}((\alpha_s/M_{SUSY})^n(\mu\tan\beta)^mA_b^{n-m})$} resummed in the
factor \DUrole{raw-tex}{$\Delta_b$}  which has a strong impact on  MSSM bottom Yukawa coupling \DUrole{raw-tex}{$g^{MSSM}(h/H/A,b\bar{b})$}as it follows from \DUrole{raw-tex}{$(\ref{eq81)$}.
This resummation defines the effective Lagrangian formalism \DUrole{raw-tex}{\cite{29hepph0511023v1}}, \DUrole{raw-tex}{\cite{hepph0511023v1}}, \DUrole{raw-tex}{\cite{hepph0603112}},
\DUrole{raw-tex}{\cite{hepph0305101v2}}

\begin{flalign}\label{eq81}
 g^{MSSM}(h, b\bar{b}) = -g^{SM}(H,b\bar{b}) \frac{1}{1+\Delta_b}
 \left [ \frac{\sin \alpha}{\cos\beta} -\Delta_b \frac{\cos\alpha}{\sin\beta}\right ],\nonumber \\
  g^{MSSM}(H, b\bar{b}) = g^{SM}(H,b\bar{b}) \frac{1}{1+\Delta_b}
 \left [ \frac{\cos \alpha}{\cos\beta} +\Delta_b \frac{\sin\alpha}{\sin\beta}\right ], \nonumber \\
   g^{MSSM}(A, b\bar{b}) = g^{SM}(H,b\bar{b}) \frac{1}{1+\Delta_b} \tan\beta
\end{flalign}

\DUrole{raw-tex}{\noindent}
The definitions \DUrole{raw-tex}{$(\ref{eq81})$} were used to in the calculation \DUrole{raw-tex}{$(\ref{eq80})$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.500000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/YRHXS_MSSM_neutral_fig2b.png}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.500000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/YRHXS_MSSM_neutral_fig2c.png}}
\caption{Total gluon-fusion cross sections of the light scalar (CP-even) MSSM Higgs boson h for two values of
\DUrole{raw-tex}{$\tan\beta=10,50$} within the \DUrole{raw-tex}{$\m_h^{max}$} scenario for \DUrole{raw-tex}{$\sqrt{s}=7$}TeV using
MSTW2008 PDFs \DUrole{raw-tex}{\cite{41,44hepph1101.0593}}. \DUrole{raw-tex}{\label{fig30}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The MSSM cross section in \DUrole{raw-tex}{$m^{max}_h$} scenario for \DUrole{raw-tex}{$M_h=120$}GeV and \DUrole{raw-tex}{$\tan\beta=30$} at
\DUrole{raw-tex}{$\sqrt{S}=7$}TeV is about one order of magnitude  larger than SM cross section at \DUrole{raw-tex}{$\sqrt{S}=14$}TeV.
Thus LHC at 7 TeV has the much larger  physics potential for
MSSM Higgs boson discovery produced in the gluon fusion process  if one  compare
with the same production of  SM Higgs boson  at
14 TeV  even when \DUrole{raw-tex}{$gg$} parton luminosity for 14 TeV is 5 times  larger (the process will occur
5 times copiously per pb`\$\textasciicircum{}\{-1\}\$` of integrated luminocity) than in the 7 TeV case
\DUrole{raw-tex}{\cite{hepph0908.3660v2}}.

For the large \DUrole{raw-tex}{$\tan\beta$} Higgs boson radiation off bottom quarks , \DUrole{raw-tex}{$gg \rightarrow bb+h/H/A+X$} becomes
the dominant Higgs boson production  process in MSSM. The inclusive total cross section of the process
in SM has been calculated in the two different approaches. Considering the mass of b quark to be large compared
to the QCD scale \DUrole{raw-tex}{$m_b \gg \Lambda_{QCD}$}, the production of the Higgs boson  associated  with b quarks
is a perturbative process which can be calculated order by order. The LO calculation ih this scheme, 4FS (4 flavor scheme)
can be found in `cite\{247,248hepph0503172v2\}`, \DUrole{raw-tex}{\cite{hepph0503117v2}}. The LO cross section of
\DUrole{raw-tex}{$gg\rightarrow bb+H$} varies strongly with the mass of the Higgs boson \DUrole{raw-tex}{$M_H$} and it is of the order of
1pb for small masses \DUrole{raw-tex}{$M_H\sim 100$}GeV  and is dropping by more than one order of magnitude when \DUrole{raw-tex}{$M_H$}
increases  to 250 GeV. The calculation of the NLO correction to the cross section of \DUrole{raw-tex}{$bbH$} production
in 4FS has been  given in \DUrole{raw-tex}{\cite{290,291hepph0503172v2}}, \DUrole{raw-tex}{\cite{hepph0503172v2}},
\DUrole{raw-tex}{\cite{4,5hepph1112.3478v1}}. However, the NLO correction
turns out to be large \DUrole{raw-tex}{\cite{hepph0304035v2}}, \DUrole{raw-tex}{\cite{hepph0503172v2}}.Because of the small b quark
mass, \DUrole{raw-tex}{$m_b$} the cross section develops large logarithms \DUrole{raw-tex}{$$ l_b = Ln(m^2_b/\mu_R), $$} where renormalization
factor \DUrole{raw-tex}{$\mu_R$} is of the order of \DUrole{raw-tex}{$M_H$\ . More precisely, every on-shell gluon that splits
into `$b\bar{b}$} pair generates one power of that logarithm. The \DUrole{raw-tex}{$K^{NLO}$} factor
\DUrole{raw-tex}{$$K^{NLO}=\sigma^{NLO}_{4FS}/\sigma^{LO}_{4FS}$$} is roughly 2 for  large values of \DUrole{raw-tex}{$M_H$} at
\DUrole{raw-tex}{$\sqrt{S}=14$}TeV  as one can
see from the Figure \DUrole{raw-tex}{$\ref{fig31}$}.
NLO QCD correction can be partly absorbed by choosing  a
low value for the factorization and remormalization scales, \DUrole{raw-tex}{$\mu_R=\mu_F\sim 1/4(M_H+2m_b)$}
\DUrole{raw-tex}{\cite{384,385hepph0503172v2}}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.500000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/lhc_2b_mh.png}}
\caption{Total inclusive cross section for \DUrole{raw-tex}{$gg->b\bar{b}+H$} at the LHC at  \DUrole{raw-tex}{$\sqrt{S}=14$}TeV as a
function of \DUrole{raw-tex}{$M_H$} with the factorization and renormalization scales set to
\DUrole{raw-tex}{$\mu_R=\mu_F\sim 1/4(M_H+2m_b)$} where CTEQ6 PDF are adopted \DUrole{raw-tex}{\cite{290hepph0503172v2}}. \DUrole{raw-tex}{\label{fig31}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Because \DUrole{raw-tex}{$\alpha_s l_b$} is rather large, it spoils the convergence of the perturbative series. The terms \DUrole{raw-tex}{$\sim l_b$}
can be summed to all orders in perturbation theory by introducing  bottom parton density functions
\DUrole{raw-tex}{\cite{hepph0204093v2}},
\DUrole{raw-tex}{$$ b(\mu,x) \sim \alpha_s(\mu)ln(m_b/\mu)$$}
which is not independent function determined by fitting to the hadronic data. The function \DUrole{raw-tex}{$b(\mu,x)$}
is generated from the pdf of light partons in the DGLAP evolution \DUrole{raw-tex}{\cite{hepph0111358v1}}.
The introduction of \DUrole{raw-tex}{$b(\mu,x)$} defines five-flavor scheme (5FS). If one could take all orders in the
perturbation theory, the 4FS and 5FS would be identical \DUrole{raw-tex}{\cite{heph1112.3478v1}}. At any finite order, the
two schemes include different parts of the all-order result, and thus the cross sections do not match
exactly. This is shown in the Figure \DUrole{raw-tex}{\ref{fig32}}.    The 5FS cross section has been calculated at NNLO
accuracy \DUrole{raw-tex}{\cite{6hepph112.3478}}, \DUrole{raw-tex}{\cite{hepph0304035v2}}, \DUrole{raw-tex}{\cite{hepph0204093v2}}.  The Figure
\DUrole{raw-tex}{\ref{fig32}} displays good agreement of the 5FS and 4FS results for smaller Higgs masses while for large
Higgs-boson masses the 5FS cross sections are  considerably large than the corresponding 4FS results. The
4FS and 5FS cross  sections at central values of MSTW2008 pdfs \DUrole{raw-tex}{\cite{41,44hepph1101.0593}}  differ by up to
30\%.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.500000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/YRHXS_MSSM_neutral_fig5a.png}}
\caption{Total production cross  sections of \DUrole{raw-tex}{$pp\rightarrow b\bar{b}H+X$} for \DUrole{raw-tex}{$\sqrt{S}=7$}TeV within the 5FS and the 4FS using MSTW2008 PDFs. The upper bands (blue bands) exhibit the combined
scale and 68\% CL PDF  \DUrole{raw-tex}{$+\alpha_s$} uncertainties of the 5FS, while the lower bands (red  bands)
include the scale uncertainties of the 4FS only.
\DUrole{raw-tex}{\label{fig32}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
In the Figure \DUrole{raw-tex}{$\ref{fig33}$} the central predictions for the gluon-fusion processes \DUrole{raw-tex}{$gg\rightarrow h/H/A$} and
the radiation off bottom quarks \DUrole{raw-tex}{$gg\rightarrow b\bar{b}+h/H/A$} within 5FS are shown as functions of the corresponding Higgs masses in
\DUrole{raw-tex}{$m_h^{max}$} scenario for values of \DUrole{raw-tex}{$\tan\beta=5,30$} at \DUrole{raw-tex}{$\sqrt{S}=7$}TeV. It can be seen from the Figure \DUrole{raw-tex}{$\ref{fig33}$}  that gluon-fusion process is dominant for \DUrole{raw-tex}{$\tan\beta=5$} while
the Higgs-boson radiation off bottom quarka plays the  more important role for   \DUrole{raw-tex}{$\tan\beta=30$}.
The mass degeneracy between CP-odd A  and two CP-even h/H     Higgs bosons doubles the cross sections of both
processes at large \DUrole{raw-tex}{$\tan\beta$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/YRHXS_MSSM_neutral_fig6a.png}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/YRHXS_MSSM_neutral_fig6b.png}}
\caption{Predictions for the  total MSSM production cross sections via gluon fusion and Higgs radiation off
bottom quarks within the 5FS for \DUrole{raw-tex}{$\sqrt{s}=7$}TeV  using NNLO and NLO
MSTW2008 PDFs  \DUrole{raw-tex}{\cite{41,44hepph1101.0593}} for the \DUrole{raw-tex}{$\m_h^{max}$} scenario at \DUrole{raw-tex}{$\tanb=5$} and \DUrole{raw-tex}{$\tanb=30$}.
\DUrole{raw-tex}{\label{fig33}}}
\end{figure}


\subsection{4.2~~~Neutral MSSM Higgs boson decays%
  \label{neutral-mssm-higgs-boson-decays}%
}

The partial decay widths of  Higgs boson decays can be significantly  affected by      large radiative  SYSY-QCD
corrections as results of the introduced resummation factor in \DUrole{raw-tex}{$(\ref{eq81})$}. Of particular interest in MSSM Higgs decays is the study of the decay rates  in the large
\DUrole{raw-tex}{$\tan\beta$} regime. One could recall that in the regime the neutral MSSM Higgs bosons are mainly decayed into
bottom quarks and tau-leptons as it follows from the tree-level partial width of the Higgs boson decay into fermion
pairs given in \DUrole{raw-tex}{\cite{111,145hepph0503172v2}} and \DUrole{raw-tex}{\cite{hepph0305101v2}}

\begin{flalign}\label{eq82}
 \Gamma_{Born}(H/h\rightarrow f\bar{f}) = \frac{G_{F}N_cM_{h/H}}{4\sqrt{2}\pi}g^2(h/H,f\bar{f})m_f^2\beta_f^3,\nonumber \\
 \Gamma_{Born}(A\rightarrow f\bar{f}) = \frac{G_{F}N_cM_{A}}{4\sqrt{2}\pi}g^2(A,f\bar{f})m_f^2\beta_f,
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$G_F$} is Fermi constant introduced in \DUrole{raw-tex}{$(\ref{eq68})$}, and \DUrole{raw-tex}{$\beta_f$} is the velocity of the fermions in
the final state \DUrole{raw-tex}{$$\beta_f=\sqrt{1-4m_f^2/M_H^2} $$} and \DUrole{raw-tex}{$N_c$} is the color factor, \DUrole{raw-tex}{$N_c =3$} for the quarks and
\DUrole{raw-tex}{$N_c=1$} for the leptons. The couplings \DUrole{raw-tex}{$g(H/h/A,f\bar{f})$} are defined  in the Table \DUrole{raw-tex}{$\ref{tbl3}$} for MSSM case
and they are equaled to 1 for SM decays.

The  partial decay width \DUrole{raw-tex}{$\Gamma(H/h/A\rightarrow f\bar{f})$}  at NNLO QCD (two-loop SM  contributions)
accuracy can be cast   to the form \DUrole{raw-tex}{\cite{hepph0305101v2}}

\begin{flalign}\label{eq83}
 \Gamma_{NLO}(H/h/A\rightarrow f\bar{f}) = \frac{G_{F}N_cM_{h/H/A}}{4\sqrt{2}\pi}g^2(h/H/A,f\bar{f})\bar{m}_f^2 [ \Delta^{NNNLO}_{QCD}+\Delta_t^{h/H/A,NNLO} ],
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$\bar{m}_f$} is the  \DUrole{raw-tex}{$\bar{MS}$} running fermion mass,  \DUrole{raw-tex}{$\Delta^{NNNLO}_{QCD}$} is the NNNLO QCD
correction  and \DUrole{raw-tex}{$\Delta_t^{NNLO,h/H/A}$} is the top quark induced contribution.
The couplings \DUrole{raw-tex}{$g(h/H/A,b\bar{b})$} affected by SUSY-QCD loops should be replaced
by their resummed versions \DUrole{raw-tex}{$(\ref{eq81})$}  to accumulate the full SUSY radiative correction in the decay rate

The results for the partial decay widths   \DUrole{raw-tex}{$\Gamma_{NLO}(h\rightarrow b\bar{b})$}  and
\DUrole{raw-tex}{$\Gamma_{NLO}(A\rightarrow b\bar{b})$}  for the 'small \DUrole{raw-tex}{$\alpha_{eff}$}' scenario
\DUrole{raw-tex}{\cite{19hepphhepph0305101v2}}  are shown in the Figure  \DUrole{raw-tex}{$\ref{fig34}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.600000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/gaml.png}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.600000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/gama.png}}
\caption{Partial decay widths of the light scalar, \DUrole{raw-tex}{$\Gamma(h\rightarrow b\bar{b})$} and the pseudoscalar,
\DUrole{raw-tex}{$\Gamma(A\rightarrow b\bar{b})$}  in the 'small \DUrole{raw-tex}{$\alpha_{eff}$} '  scenario. The shaded bands
reflects uncertainties due to the chose of the scale in the strong coupling constant \DUrole{raw-tex}{$\alpha_s$}
\DUrole{raw-tex}{\label{fig34}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The region of \DUrole{raw-tex}{$M_A\sim 150$}GeV
corresponds to the vanishing Yukawa coupling
\DUrole{raw-tex}{$$g(h,b\bar{b})=-\frac{\sin\alpha}{\cos\beta}\frac{1}{1+\Delta_b}\left(1-\frac{\Delta_b}{\tan\alpha\tan\beta}\right)
\rightarrow 0,\,\,\ \, \tan\beta\tan\alpha\rightarrow\Delta_b.$$}
Branching ratios \DUrole{raw-tex}{$\Gamma(h/A\rightarrow b\bar{b},\tau\bar{\tau},gg,t\bar{t})/\Gamma_{Total}$}  of the light scalar \DUrole{raw-tex}{$h$}
and the pseudoscalar \DUrole{raw-tex}{$A$} are illustrated in the Figure \DUrole{raw-tex}{$\ref{fig35}$}.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.600000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/brl.png}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.600000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/bra.png}}
\caption{Branching ratios  of the light scalar, \DUrole{raw-tex}{$Br(\rightarrow b\bar{b})$} and the pseudoscalar,
\DUrole{raw-tex}{$Br(A\rightarrow b\bar{b})$}  in the 'small \DUrole{raw-tex}{$\alpha_{eff}$} '  scenario. The shaded bands
reflects uncertainties due to the chose of the scale in the strong coupling constant \DUrole{raw-tex}{$\alpha_s$}
\DUrole{raw-tex}{\label{fig35}}}
\end{figure}


\subsection{4.3~~~Constraints on the mass of the  Higgs boson  from direct searches and indirect precise measurements%
  \label{constraints-on-the-mass-of-the-higgs-boson-from-direct-searches-and-indirect-precise-measurements}%
}


\subsubsection{4.3.1~~~Theoretical constraints on the Higgs Boson mass%
  \label{theoretical-constraints-on-the-higgs-boson-mass}%
}

One of the main interests in High Energy physics is the search for
evidence of the Higgs boson and the determination of its mass.  In the SM, the Higgs boson is
given by \DUrole{raw-tex}{$m_H=\sqrt{\lambda/2}v$}, where \DUrole{raw-tex}{$\lambda$} is the Higgs self-coupling parameter and
\DUrole{raw-tex}{$v$} is the vacum expectation value of the Higgs field. Since \DUrole{raw-tex}{$\lambda$} is unknown, the value of the
SM Higgs boson mass \DUrole{raw-tex}{$m_H$} cannot be predicted.
There is an upper bound on the Higgs boson mass  based on
the perturbativity of the theory up to the scale \DUrole{raw-tex}{$\Lambda$} at which the
SM breaks down. This upper bound is connected to an unsatisfactory high energy
behaviour of the Higgs quartic self-coupling \DUrole{raw-tex}{$\lambda$}.  If \DUrole{raw-tex}{$m_H$} is
too large, then the Higgs self-coupling
diverges at some scale \DUrole{raw-tex}{$\Lambda$} below the Planck scale

% cite1

% Thomas Hambye and Kurt Riesselmann.Matching conditions and Higgs mass upper bounds revisited.

% Phys. Rev., D55:7255-7262, 1997.

\DUrole{raw-tex}{\cite{cite1,hepph9610272}}
The lower limit on \DUrole{raw-tex}{$M_H$} is determined by the possible instability of the Higgs potential \DUrole{raw-tex}{$V(H)$}.
The quantum loop corrections make the derivative  \DUrole{raw-tex}{$V^{\prime}(H)$} to be negative at the large  magnitude of
the scalar field \DUrole{raw-tex}{$H$} and the potential would become unbound from below \DUrole{raw-tex}{\cite{cite2,hepph9606386v}}.

% cite2

% Guido Altarelli (CERN), G. Isidori (INFN, Rome & Rome U.). Jul 1994. 8 pp.

% Published in Phys.Lett. B337 (1994) 141-144

Fixing the top quark mass to be 175 GeV, using the two-loop  beta function   for the  self-coupling
\DUrole{raw-tex}{$\lambda$}, the  upper bound on \DUrole{raw-tex}{$m_H$} is depicted in the Figure \DUrole{raw-tex}{$\ref{fig36}$}. The uncertainty
connected to the upper bound is caused by variations of the matching scale  and the cutoff conditions which
determines the validity of the perturbative approximation \DUrole{raw-tex}{\cite{cite1}}.
The additional experimental uncertainty due
to the top quark mass is represented by the cross-hatched area in the Figure \DUrole{raw-tex}{$\ref{fig36}$}.
The lower bound on \DUrole{raw-tex}{$m_H$}  is shown  as well. At the large scale \DUrole{raw-tex}{$\Lambda$}, the stability bound is
approximated by requiring the Higgs running coupling to remain positive, \DUrole{raw-tex}{$\lambda(\Lambda)>0$}
\DUrole{raw-tex}{\cite{1 cite2}}. Such  analysis has been carried out at the two-loop level \DUrole{raw-tex}{\cite{cite2}}
The solid area of the lower bound indicates the theoretical uncertainty from  hight order effects.

% figure fig36  goes here from cite1
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.600000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/figfinal.png}}
\caption{The upper solid area indicates the sum of theoretical
uncertainties in the \DUrole{raw-tex}{$M_H$} upper bound when keeping \DUrole{raw-tex}{$m_t=175$} GeV
fixed. The cross-hatched
area shows the additional uncertainty when varying \DUrole{raw-tex}{$m_t$} from 150 to
200 GeV.
The upper edge corresponds to Higgs
masses for which the SM Higgs sector ceases to be
meaningful at scale \DUrole{raw-tex}{$\Lambda$}, and the
lower edge indicates a value of
\DUrole{raw-tex}{$m_H$} for which perturbation theory is certainly expected to be
reliable at scale \DUrole{raw-tex}{$\Lambda`$\ .
The lower solid area represents the theoretical uncertaintites in
the `$m_H$} lower bounds derived from stability requirements
using \DUrole{raw-tex}{$m_t=175$} GeV and     \DUrole{raw-tex}{$\alpha_s=0.118`$\  `\cite{cite1}}. \DUrole{raw-tex}{\label{fig36}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The Higgs boson mass \DUrole{raw-tex}{$m_H$} can provide an important constraint on the scale up to which
the SM remains successful as an effective theory. As it is shown in the Figure \DUrole{raw-tex}{$\ref{fig36}$},
that the effective SM would survive all the way to the Plank scale \DUrole{raw-tex}{$M_{Pl}$\  if the Higgs boson mass
would lie within the range `130 GeV $\lesssim m_H \lesssim$ 180 GeV}.
The Figure \DUrole{raw-tex}{$\ref{fig37}$} \DUrole{raw-tex}{\cite{cite3,hepph1112.3022}}  shows the RG evolution of the self-coupling \DUrole{raw-tex}{$\lambda$}

% cite3

% Joan Elias-Miro Phys.Lett. B709 (2012) 222-228

for \DUrole{raw-tex}{$m_H=124$}GeV in the \DUrole{raw-tex}{$\bar{MS}$} scheme. The coupling \DUrole{raw-tex}{$\lambda$} becomes  negative
around 10`\$\textasciicircum{}9\$` GeV. The scale can be shifted by a few orders of magnitude  by varying the top quark
mass \DUrole{raw-tex}{$m_t$}. The new deeper minimum  of the Higgs potential  at the negative  \DUrole{raw-tex}{$\lambda$}
could be the metastable electroweak vacum which lifetime is longer than the age of the Universe
\DUrole{raw-tex}{\cite{cite3,hepph1112.3022}}. The  meta-stability with quantum tunneling at zero
temperature is the region above  the shaded area in red  in the Figure \DUrole{raw-tex}{$\ref{fig37}$}.

% figure fig37 goes here form cite3
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=1.100000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/run124.pdf}}
\caption{RG evolution of the Higgs self coupling, for different Higgs masses
for the central value of \DUrole{raw-tex}{$m_t$} and \DUrole{raw-tex}{$\alpha_s$} , as well as for
\DUrole{raw-tex}{$\pm 2\sigma$} variations of \DUrole{raw-tex}{$m_t$} (dashed lines) and  \DUrole{raw-tex}{$\alpha_s$} (dotted lines).
For negative values of \DUrole{raw-tex}{$\lambda$} ,
the life-time of the SM vacuum due to quantum tunneling at zero temperature
is longer than the age of the Universe as long as \DUrole{raw-tex}{$\lambda`$ remains
above the region shaded in red, which takes into account the finite corrections to the
effective bounce action renormalised at the same scale as `$\lambda$} \DUrole{raw-tex}{\cite{cite3,hepph1112.3022}}
\DUrole{raw-tex}{\label{fig37}}}
\end{figure}


\subsubsection{4.3.2~~~Indirect constraints on the neutral MSSM Higgs Boson mass%
  \label{indirect-constraints-on-the-neutral-mssm-higgs-boson-mass}%
}

Indirect experimental bounds for the Higgs boson mass
are obtained from the global fit to precision measurements of electroweak
observables \DUrole{raw-tex}{\cite{cite4}}.

% cite4

% gFitter

% http://project-gfitter.web.cern.ch/project-gfitter/Standard_Model/

% hepph http://arxiv.org/pdf/1209.2716v2.pdf

% http://arxiv.org/pdf/1107.0975v2.pdf

% mastercode

% http://www.itp.uzh.ch/events/darkattack/talks/Buchmueller.pdf

% http://mastercode.web.cern.ch/mastercode/talks/2011/EPS-2011-samr.pdf

% http://arxiv.org/pdf/1112.3564v2.pdf

% http://arxiv.org/pdf/0808.4128v1.pdf

% http://mastercode.web.cern.ch/mastercode/results.php

% Fittino

% hepph 1204.4199v1.pdf

% http://www-flc.desy.de/fittino/fittino_publications.html

% http://arxiv.org/pdf/0907.2589v1.pdf

The input to the global fit consists
on the set of 'Low energy' observables which are confronted with MSSM parameter space.
While no direct evidence for SUSY particles has been
found to date, these particles contribute to higher order
corrections to measured physical observables in a welldefined
and calculable way if SUSY is realised in Nature
The MSSM Higgs boson contributes to the \DUrole{raw-tex}{$W^{\pm}$} and \DUrole{raw-tex}{$Z$}
vacuum polarization through loop effects, leading to a logarithmic
sensitivity of the ratio of the \DUrole{raw-tex}{$W^{\pm}$} and \DUrole{raw-tex}{$Z$} gauge boson
masses over the Higgs boson mass.
Low energy observables   can be grouped in four classes:
%
\begin{itemize}

\item Rare decays of B- and K-mesons

\item The anomalous magnetic of the muon

\item Precision measurements of electroweak physics and the Higgs boson mass limits from high energy colliders: LEP,Tevatron and LHC

\item The relic density of cold dark matter in the Universe

\end{itemize}

All the exploited measurements and their values were summarised in  \DUrole{raw-tex}{\cite{hepph0907.2589v1,hepph1204.4199}}.
The global \DUrole{raw-tex}{$\chi^2$} could be  calculated in three steps.
%
\begin{itemize}

\item First, for all input measurements \DUrole{raw-tex}{$\chi^2_{\rm meas}$}

\end{itemize}

\begin{equation}\label{eq84}
\chi^2_{\rm meas}=\sum_{i=1}^{N_{\rm meas}}\left(\frac{O_{\rm meas}^i
-O_{\rm pred}^i (\vec{P})}{\sigma^{i}}\right)^2
\end{equation}
%
\begin{description}
\item[{\DUrole{raw-tex}{\noindent}}] \leavevmode 
is calculated for each parameter point \DUrole{raw-tex}{$\vec{P}$} where the sum runs

\end{description}

over all \DUrole{raw-tex}{$N_{\mathrm{meas}}$} measurements.
%
\begin{itemize}

\item In case of upper bounds (e.g., the bounds on \DUrole{raw-tex}{${\cal B}(B_s\to\mu\mu)$} )
the \DUrole{raw-tex}{$\chi^2_{\rm meas+bound}$} is added to the global \DUrole{raw-tex}{$\chi^2$}

\end{itemize}

\begin{equation}\label{eq85}
\chi^2_{\rm bound}=\sum_{i=1}^{N_{\rm bound}}
\left\{
\begin{array}{cl}
\left(\frac{O_{\rm limit}^i-O_{\rm pred}^i(\vec{P})}{\sigma^{i}}\right)^2
 & \mathrm{for}\,O_{\rm pred}^i(\vec{P})>O_{\rm limit}^i\\[3mm] 0 &
 \mathrm{otherwise}\end{array}
\right.
\end{equation}
%
\begin{description}
\item[{\DUrole{raw-tex}{\noindent}}] \leavevmode 
where \DUrole{raw-tex}{$\sigma^i$} is the assumed theoretical uncertainty of the prediction.

\end{description}
%
\begin{itemize}

\item Finally, the \DUrole{raw-tex}{$\chi^2_{\rm Higgs,SUSY,Dark}$} contributions from the  Higgs searches at Tevatron and LHC,
from the LHC SUSY search constraints, and from the
direct and indirect detection of the  dark matter  are
calculated and added to the    global \DUrole{raw-tex}{$\chi^2$}.

\end{itemize}

\DUrole{raw-tex}{\noindent}
The input predictions of MSSM for the global fit \DUrole{raw-tex}{$\chi^2$} are obtained in the
framework of the constrained MSSM (the mSUGRA model), \textbf{cMSSM},  \DUrole{raw-tex}{\cite{43,45hepph 1204.4199}}
which has

\begin{equation}\label{eq86}
 M_0,\,M_{1/2},\,A_0,\,\tan\beta,\,\mathrm{sgn}(\mu)\,.
 \end{equation}

\DUrole{raw-tex}{\noindent}
only 5 free parameters beyond SM.
The pull plots of the fit based on only
'Low Energy Observables'  ( \textbf{LEO} ),  \DUrole{raw-tex}{$\chi^2_{\rm meas} + \chi^2_{\rm bound}$}  and of the \textbf{LEO} fit
with the inclusion of
the limits on the SM Higgs boson mass obtained in  ATLAS and CMS experiments, \DUrole{raw-tex}{$\chi^2{Higgs}$},
\DUrole{raw-tex}{\cite{62,63hepph1204.4199}} are show in the Figure \DUrole{raw-tex}{$\ref{fig38}$}.

% figure fig38 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.300000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Fittino_pulls.png}}
\caption{Contribution of the individual observables into the global fit. The \textbf{LEO} fit (on the left side)
and \textbf{LEO}  fit with the Higgs mass constraints from LHC (on the right side ) are illustrated.
The numbers in the second and third collumns correspond to \DUrole{raw-tex}{$O_{\rm meas}^i$}
and \DUrole{raw-tex}{$O_{\rm pred}^i(\vec{P}_{fitted})$} respectively \DUrole{raw-tex}{\label{fig38}} \DUrole{raw-tex}{\cite{hepph1204.4199}}.}
\end{figure}

\DUrole{raw-tex}{\noindent}
From the pull plot in the Figure \DUrole{raw-tex}{$\ref{fig38}$} one can deduce that the LHC limits push the fit
into a region where, in particular, \DUrole{raw-tex}{$a_{\mu}$} is not described very well anymore.
The high sparticle masses required to yield  the Higgs mass around 126 GeV \DUrole{raw-tex}{$$m_h=126\pm 0.5\,\,GeV $$}
push the best fit point into the  region  of large \DUrole{raw-tex}{$M_0$}. This leads to the large value of
\DUrole{raw-tex}{$\chi^2/ndf$} which is 18.4/9  for the right plot shown in the Figure \DUrole{raw-tex}{$\ref{fig38}$}.This tells
that    \DUrole{raw-tex}{$m_h=126\,\,GeV $}   is hardly compatible with \textbf{cMSSM}.
The predicted range of sparticles   and Higgs bosons masses are also estimated in the global fit \DUrole{raw-tex}{$\chi^2$}.
The flat profile of \textbf{LEO} fit, \DUrole{raw-tex}{$\chi^2_{\rm meas}+\chi^2_{\rm bound}$},
\DUrole{raw-tex}{\cite{hepph1204.4199}} for  \DUrole{raw-tex}{$M_0$} and
\DUrole{raw-tex}{$M_{1/2}$} parameters of \textbf{cMSSM}  predicts sparticles at the masses \DUrole{raw-tex}{\sim 1$}TeV with \DUrole{raw-tex}{$1\sigma$} bands
equaled roughly to 300 GeV. This is shown in the Figure \DUrole{raw-tex}{$\ref{fig39}$}.

% figure fig39 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Fittino_sparticles.png}}
\caption{Predicted distribution of sparticle and Higgs boson masses from  \textbf{LEO} fit.
The full uncertainty band gives the 1-dimensional \DUrole{raw-tex}{$2\sigma$} uncertainty of each mass defined
by the region \DUrole{raw-tex}{$\Delta\chi^2 < 4$} after profiling over all hidden dimensions \DUrole{raw-tex}{\label{fig39}} \DUrole{raw-tex}{\cite{hepph1204.4199}}.}
\end{figure}

\DUrole{raw-tex}{\noindent}
The only particle with a very strong
bound on its mass is the lightest Higgs boson, \DUrole{raw-tex}{$h$},  as one can see from the Figure \DUrole{raw-tex}{$\ref{fig39}$}\DUrole{raw-tex}{$$ m_h = (117 \pm 1.5)\,\,\, GeV. $$}
A very strong constraint on  \textbf{cMSSM} is expected from a direct measurement of the
lightest Higgs boson mass \DUrole{raw-tex}{$m_h$}. This is explicitly shown  in the Figure \DUrole{raw-tex}{$\ref{fig40}$},
where the profiles  for
\DUrole{raw-tex}{$m_h$} are displayed:
%
\begin{itemize}

\item for  \DUrole{raw-tex}{$\chi^2_{\rm meas}+\chi^2_{\rm bound}$},
the LHC fit  (without direct constraint on the Higgs mass);

\item for  \DUrole{raw-tex}{$\chi^2_{\rm meas}+\chi^2_{\rm bound}+\chi^2{\rm Higgs}$},
the LHC fit with constraints on \DUrole{raw-tex}{$m_h$} which are \DUrole{raw-tex}{$$m_h =126\pm 2 \pm 3\,\,\,GeV,$$} at the fixed top mass, \DUrole{raw-tex}{$m_t$};

\item for  the same  LHC fit but with floating \DUrole{raw-tex}{$m_t$} when the top quark mass changes in the range  \DUrole{raw-tex}{$$m_t= 173.2\pm1.34\,\,\,GeV.$$}

\end{itemize}

% figure fig40 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Fittino_higgs.png}}
\caption{The dependence of the minimal \DUrole{raw-tex}{$\chi^2$} of the fit on \DUrole{raw-tex}{$m_h$} for different input observable sets
for the \textbf{cMSSM}with and without \DUrole{raw-tex}{$m_h = (126\pm 2\pm 3)GeV$}  constraint \DUrole{raw-tex}{\label{fig40}}  \DUrole{raw-tex}{\cite{hepph1204.4199}}.}
\end{figure}

\DUrole{raw-tex}{\noindent}
It can  be seen that the \textbf{cMSSM} fits ( solid brown  curves) can barely incorporate \DUrole{raw-tex}{$m_h = 126$}GeV.
At this edge the \DUrole{raw-tex}{$\chi^2$} rises dramatically excluding any higher Higgs boson mass values. This
limit is just slightly increased by floating \DUrole{raw-tex}{$m_t$} corresponded to  a dotted green curve in the Figure
\DUrole{raw-tex}{$\ref{fig40}$}


\subsubsection{4.3.3~~~Direct measurements of the neutral MSSM Higgs Boson mass%
  \label{direct-measurements-of-the-neutral-mssm-higgs-boson-mass}%
}

At LEP2, substantial data samples were collected
at center-of-mass energies up to 209 GeV.
The four LEP collaborations, ALEPH, DELPHI, L3 and OPAL,
have searched for the neutral Higgs
bosons which are predicted by MSSM \DUrole{raw-tex}{\cite{cite5}}.
The LEP2 data sample used by the LEP Higgs working group was of  2461 \DUrole{raw-tex}{$pb^{-1}$} integrated luminocity.

% cite5

% http://lephiggs.web.cern.ch/LEPHIGGS/papers/July2005_MSSM/index.html

% http://lephiggs.web.cern.ch/LEPHIGGS/papers/index.html

% hepph hep-ex/0602042 , Eur.Phys.J.C47:547-587,2006

% hepph P Teixeira-Dias 2008 J. Phys.: Conf. Ser. 110 042030 doi:10.1088/1742-6596/110/4/042030

\DUrole{raw-tex}{\noindent}
Data recorded at each center-of-mass energy were studied independently.
The combination of the results of the four experiments was based on the ratio of the extended
likelihoods for the signal-plus-background hypothesis and the background-only hypothesis
\DUrole{raw-tex}{\cite{Ref1 doi:10.1088/1742-6596/110/4/042030 }}
No single LEP experiment had the power to distinguish between the two hypotheses at
more than the two-sigma level. The combination of the LEP data yields a 95\% C.L. lower
bound of 114.4 GeV for the mass of the lightest MSSM Higgs boson \DUrole{raw-tex}{$H1(h)$}    as it is
show in the left plot of the Figure \DUrole{raw-tex}{$\ref{fig41}$}.  Here \DUrole{raw-tex}{$S_{95}$} is \DUrole{raw-tex}{$\frac{\sigma_{95}}{\sigma_{ref}}$},
where \DUrole{raw-tex}{$\sigma_{95}$} is the largest cross-section compatible with the data, at the 95\% CL, and
\DUrole{raw-tex}{$\sigma_{ref}$} is the reference cross section     taken to   be the Standard Model Higgs production
cross section for the Higgsstrahlung,  \DUrole{raw-tex}{$e^+e^-\rightarrow h+Z$}, process.

% fig41 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/LEP_higgs.png}}
\caption{The 95\% CL upper bound, \DUrole{raw-tex}{$S_{95}$} for the cross section of the Higgsstrahlung as a function of the
Higgs boson mass is shown in left plot. The Higgs boson is assumed to decay exclusively to \DUrole{raw-tex}{$b\bar{b}$}.
The right plot shows exclusions at 95 \% CL (light green) and 99.7\% CL (dark green) in the case of
CP-conserving \DUrole{raw-tex}{$m_h^{max}$} benchmark scenario of MSSM. A dashed line  indicates the boundary of the region
which is expected to be excluded, at 95\% CL,  on the basis of the MC study with no signal \DUrole{raw-tex}{\cite{hepex0602042 }}.
\DUrole{raw-tex}{\label{fig41}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
An excess of data  consistent with a Higgs boson of mass  \DUrole{raw-tex}{$m_h=115$}GeV was seen.
The exclusion for the \DUrole{raw-tex}{$m_h^{max}$} benchmark scenario is shown in the Figure \DUrole{raw-tex}{$\ref{fig41}$} as well.
In the region with \DUrole{raw-tex}{$\tan\beta$} less than about five, the exclusion is provided by the Higgsstrahlung
process, \DUrole{raw-tex}{$e^+e^-\rightarrow h+Z$}, giving the lower limit of about 114 GeV.
At high \DUrole{raw-tex}{$\tan\beta$}, the pair production process, \DUrole{raw-tex}{$e^+e^-\rightarrow h+H$},   is most useful, providing limits in the vicinity of
93 GeV for \DUrole{raw-tex}{$m_h$} \DUrole{raw-tex}{\cite{hepph hep-ex/0602042 }}.

At Tevatron, the CDF and D0 collaborations have searched for the neutral MSSM
Higgs bosons produced in association with bottom quarks and
which decay into \DUrole{raw-tex}{$b\bar{b}$} \DUrole{raw-tex}{\cite{226,227pdg}} and into \DUrole{raw-tex}{$\tau^+\tau^-$} \DUrole{raw-tex}{\cite{228,229pdg}}.
The most recent searches in the \DUrole{raw-tex}{$b\bar{b}+h/H/A$} channel with \DUrole{raw-tex}{$h/H/A\rightarrow b\bar{b}$}
analyze approximately 2.6 fb\DUrole{raw-tex}{$^{-1}$} of data (CDF)  and 5.2  fb\DUrole{raw-tex}{$^{-1}$}   (D0), seeking the
events with at least three b-tagged jets. The upper limits at 95\% CL
on the product of the cross section and
the branching ratio
\DUrole{raw-tex}{$$\sigma(p\bar{p}\rightarrow b\bar{b}+h/H/A)\times Br(h/H/A\rightarrow b\bar{b}), $$}
using the modified frequentist technique \DUrole{raw-tex}{\cite{44,44hepex 1207.2757}} with
Log Likelihood Ratio (\textbf{LLR}), were extracted. These limits are model-independent and presented in
the Figure \DUrole{raw-tex}{$\ref{fig42}$} (a left plot). Excesses of events above the SM background expectation are observed
for \DUrole{raw-tex}{$M_{h/H/A}=120$}GeV and \DUrole{raw-tex}{$M_{h/H/A}=140$}GeV with significance of \DUrole{raw-tex}{$\sim 2 $} standard deviations.

% fig42 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.300000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Tevatron_higgs.png}}
\caption{Model independent 95\% CL upper limits on the
\DUrole{raw-tex}{$\sigma(p\bar{p}\rightarrow b\bar{b}+h/H/A)\times Br(h/H/A\rightarrow b\bar{b})$} for the combined
CDF and D0 analyse is shown on the left side. 95\% CL lower limit in the \DUrole{raw-tex}{$(M_A,\tan\beta)$} plane
for the \DUrole{raw-tex}{$m_h^{max}, \mu=-200$}GeV benchmark scenario \DUrole{raw-tex}{\cite{54 hepex 1207.2757}} is presented on
the right. The exclusion limit obtained from the LEP experiments is shown as well. \DUrole{raw-tex}{\cite{heppex 1207.2757}}
\DUrole{raw-tex}{\label{fig42}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
These results are interpreted in the terms of \DUrole{raw-tex}{$m_h^{max}$} \DUrole{raw-tex}{\cite{54hepex 1207.2757}}
benchmark scenario with MSSM.  The exlusion limits in the \DUrole{raw-tex}{$(M_A,\tan\beta)$} plane for this scenario
are shown in the right plot of the Figure \DUrole{raw-tex}{$\ref{fig42}$}. However, because more than
half of the integrated luminosity from the Tevatron has still to be analyzed, the limits might be updated in the future.

The LHC has the potential to explore a broad range of  SUSY parameter space
through the search for non-SM-like Higgs bosons. In 2011, ATLAS and CMS   performed searches
for neutral MSSM Higgs bosons which were decaying to the pair  of tau-leptons,
\DUrole{raw-tex}{$h/H/A\rightarrow\tau^{+}\tau^-$}, in the \DUrole{raw-tex}{$pp$} collisions at \DUrole{raw-tex}{$\sqrt{S}=7$}TeV. Published
ATLAS \DUrole{raw-tex}{\cite{234,235pdg}} and CMS \DUrole{raw-tex}{\cite{138,139pdf}} results were obtained in \DUrole{raw-tex}{1.06}fb\DUrole{raw-tex}{$^{-1}$}  and
4.6 fb\DUrole{raw-tex}{$^{-1}$} of data respectevely.  The searches were performed in categories of
the decays of the two tau leptons: \DUrole{raw-tex}{$$e\tau_{had},\,\, \mu\tau_{had},\,\, e\mu,\,\,  \mu\mu,$$}
where \DUrole{raw-tex}{$\tau_{had}$} denotes a tau-lepton which decays to one or more hadrons plus a tau neutrino,
\DUrole{raw-tex}{$e$} means \DUrole{raw-tex}{$\tau\rightarrow e\nu_e\nu_{\tau}$}  and \DUrole{raw-tex}{$\nu$} corresponds to the case
\DUrole{raw-tex}{$\tau\rightarrow \mu\nu_{\mu}\nu_{\tau}$}. Obtained constraints from ATLAS and CMS searches for this channel
are shown in the Figure \DUrole{raw-tex}{$\ref{fig43}$}.

% fig43 goes here
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[scale=0.400000]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/LHC_higgs.png}}
\caption{The 95\% CL exlusion contours in  the \DUrole{raw-tex}{$m_h^{max},\mu=+200$}GeV  MSSM scenario obtained
by the ATLAS \DUrole{raw-tex}{\cite{234pfg}}, CMS \DUrole{raw-tex}{\cite{138pdg}}  and D0 \DUrole{raw-tex}{\cite{232pdg}} collaborations.
Also the region excluded by LEP \DUrole{raw-tex}{\cite{34pdg}} is shown. \DUrole{raw-tex}{\label{fig43}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The Figure \DUrole{raw-tex}{$\ref{fig43}$}  presents a broad region with intermediate
\DUrole{raw-tex}{$\tan\beta$} and large values of \DUrole{raw-tex}{$M_A$} that is not tested  by the searches of
the  neutral or charged Higgs boson from other experiments, and which
might be difficult to cover completely via these experiments, even with much larger data sets.

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